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

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

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

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

p2 in {p1,p2} by TARSKI:def 2;

then p2 in rng <*p1,p2*> by Lm1;

hence <*p1,p2*> :- p2 = <*p2*> ^ (<*p1,p2*> |-- p2) by Th41

.= <*p2*> ^ {} by A1, Th33

.= <*p2*> by FINSEQ_1:34 ;

:: thesis: verum

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

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

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

p2 in {p1,p2} by TARSKI:def 2;

then p2 in rng <*p1,p2*> by Lm1;

hence <*p1,p2*> :- p2 = <*p2*> ^ (<*p1,p2*> |-- p2) by Th41

.= <*p2*> ^ {} by A1, Th33

.= <*p2*> by FINSEQ_1:34 ;

:: thesis: verum