let D be non empty set ; :: thesis: for p1, p2 being Element of D holds Rotate <*p1,p2*>,p2 = <*p2,p2*>
let p1, p2 be Element of D; :: thesis: Rotate <*p1,p2*>,p2 = <*p2,p2*>

per cases ( p1 = p2 or p1 <> p2 ) ;

suppose p1 = p2 ; :: thesis: Rotate <*p1,p2*>,p2 = <*p2,p2*>

hence Rotate <*p1,p2*>,p2 = <*p2,p2*> by Th101; :: thesis: verum

end;

suppose A1: p1 <> p2 ; :: thesis: Rotate <*p1,p2*>,p2 = <*p2,p2*>

rng <*p1,p2*> = {p1,p2} by Lm1;
then p2 in rng <*p1,p2*> by TARSKI:def 2;
hence Rotate <*p1,p2*>,p2 = (<*p1,p2*> :- p2) ^ ((<*p1,p2*> -: p2) /^ 1) by Def2
.= <*p2*> ^ ((<*p1,p2*> -: p2) /^ 1) by A1, Th57
.= <*p2*> ^ (<*p1,p2*> /^ 1) by A1, Th53
.= <*p2*> ^ <*p2*> by Th50
.= <*p2,p2*> by FINSEQ_1:def 9 ;
:: thesis: verum

end;

end;