 Welcome back, let us try this out. Let us say that we now have three temperature levels T1, T2 and T3, all three distinct from each other. That means we have T1, T2 and T3 such that T1 is not equal to T2, T2 is not equal to T3 and again T3 is not equal to T1. When we have inequalities, these three distinct inequalities are necessary. And let us say that T1 and T2 are such that an engine, let me say E12 works between them and produces W12 which is positive. And let us say this is Q1 of engine 12, Q2 of engine 12 and T2 and T3 are such that an engine E23 works between them like this. It absorbs Q1, 23 from the reservoir at T2, it rejects Q2, 23 for the reservoir at T3 and it provides a positive amount of work W23. So, this is given such an engine works and also such an engine works. Now, the question is if we attempt an engine between T1 and T3, which way will it work? The options are A, notice it has to be T1 and T3. The options are A, the engine 13 will work by absorbing from T1 and rejecting to T3 and producing W13 or B. Option B is T1, T3, let me say this is E31. It absorbs heat from T3, rejects heat to T1 and produces W31 greater than 0. We have to decide on whether this works or this works. Is it A or is it B? Since we have already gone through such exercises, I will leave it to you to show that the first option leads to no inconsistency. We will now look at the second option. What we will do is we will lay out the three reservoirs like this, T1, T2, T3. We have one engine, I have called it E12 working between T1 and T2. We have another engine E23 working between T2 and T3. Here we have, here also we have W23 greater than 0. Now our option B said that we have an engine E31 working between T3 and T1 such that it absorbs heat from the reservoir at T3 and rejects it to the reservoir at T1 and produces a positive amount of work W31 greater than 0. Now let us rename the interactions. Let us say that the first engine E12 absorbs heat Q1 from the reservoir at T1. Let it reject Q2 to the reservoir at T2. Let us adjust this engine E23 in such a way that it now absorbs Q2 from the reservoir at T2 and rejects Q3 to the reservoir at T3. And let us adjust this engine E31 such that it absorbs Q3 from the reservoir at T3 and rejects say Q4 to the reservoir at T1. So the heat flow rates are Q1 from T1 to E12, Q2 from E12 to T2 and then to E23 and Q3 from E23 to T3 and then to E31 and finally Q4 from E31 to the reservoir at T1. Now notice that W12 is positive, E being an engine so we should have Q2 less than Q1. Again E23 is an engine so W23 is positive so we should have Q3 less than Q2 which inter is less than Q1 so Q3 is less than Q1. Then E31 has an output W31 which is also positive so you must have Q4 less than Q3. So we must have Q4 which is less than Q3 and since Q3 is less than Q2 less than Q1 finally we should have Q4 less than Q1 giving you Q1 minus Q4 to be a positive number. Now let us consider this system T1 is out of it but the reservoirs at T2, T3 and all the three engines are included in it. Now look at this, this is an engine because the work output is W12 plus W31 plus W23 each of the components is greater than 0 so this is greater than 0. What is the heat input? The heat input is Q1 minus Q4 so heat supplied is Q1 minus Q4 which by first law will turn out to be equal to W and only one reservoir T1 so this turns out to be a 1T heat engine which violates the Kelvin Planck statement and that means that going back to our earlier work this is not possible because if we assume that this is true it will violate existence of such an E31 will violate the second law of thermodynamics as mentioned by the Kelvin Planck statement. So what does this finally mean? This means that if an engine works between T1 and T2 like this and an engine works between T2 and T3 like this then if we try to create or run an engine between T1 and T3 then it will run like this absorbing from T1 rejecting to T3 and producing some work output and not like this this we can keep in brackets just to emphasize because this is just reiterating what is in the left part such a thing cannot exist and that means now we know that if we say using this that T1 is higher than T2 and T2 is higher than T3 then T1 must be higher than T3. Thank you.