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

<*p1,p2*> -: p2 = <*p1,p2*>

let p1, p2 be Element of D; :: thesis: ( p1 <> p2 implies <*p1,p2*> -: p2 = <*p1,p2*> )

assume A1: p1 <> p2 ; :: thesis: <*p1,p2*> -: p2 = <*p1,p2*>

rng <*p1,p2*> = {p1,p2} by Lm1;

then p2 in rng <*p1,p2*> by TARSKI:def 2;

hence <*p1,p2*> -: p2 = (<*p1,p2*> -| p2) ^ <*p2*> by Th40

.= <*p1,p2*> by A1, Th26 ;

:: thesis: verum

<*p1,p2*> -: p2 = <*p1,p2*>

let p1, p2 be Element of D; :: thesis: ( p1 <> p2 implies <*p1,p2*> -: p2 = <*p1,p2*> )

assume A1: p1 <> p2 ; :: thesis: <*p1,p2*> -: p2 = <*p1,p2*>

rng <*p1,p2*> = {p1,p2} by Lm1;

then p2 in rng <*p1,p2*> by TARSKI:def 2;

hence <*p1,p2*> -: p2 = (<*p1,p2*> -| p2) ^ <*p2*> by Th40

.= <*p1,p2*> by A1, Th26 ;

:: thesis: verum