let S1, S2 be BinOp of the carrier of F; :: thesis: ( ( for x, y being Element of F holds S1 . (x,y) = the addF of F . (x,((comp F) . y)) ) & ( for x, y being Element of F holds S2 . (x,y) = the addF of F . (x,((comp F) . y)) ) implies S1 = S2 )

assume that

A3: for x, y being Element of F holds S1 . (x,y) = the addF of F . (x,((comp F) . y)) and

A4: for x, y being Element of F holds S2 . (x,y) = the addF of F . (x,((comp F) . y)) ; :: thesis: S1 = S2

assume that

A3: for x, y being Element of F holds S1 . (x,y) = the addF of F . (x,((comp F) . y)) and

A4: for x, y being Element of F holds S2 . (x,y) = the addF of F . (x,((comp F) . y)) ; :: thesis: S1 = S2

now :: thesis: for p being object st p in [: the carrier of F, the carrier of F:] holds

S1 . p = S2 . p

hence
S1 = S2
by FUNCT_2:12; :: thesis: verumS1 . p = S2 . p

let p be object ; :: thesis: ( p in [: the carrier of F, the carrier of F:] implies S1 . p = S2 . p )

assume p in [: the carrier of F, the carrier of F:] ; :: thesis: S1 . p = S2 . p

then consider x, y being object such that

A5: ( x in the carrier of F & y in the carrier of F ) and

A6: p = [x,y] by ZFMISC_1:def 2;

thus S1 . p = S1 . (x,y) by A6

.= the addF of F . (x,((comp F) . y)) by A3, A5

.= S2 . (x,y) by A4, A5

.= S2 . p by A6 ; :: thesis: verum

end;assume p in [: the carrier of F, the carrier of F:] ; :: thesis: S1 . p = S2 . p

then consider x, y being object such that

A5: ( x in the carrier of F & y in the carrier of F ) and

A6: p = [x,y] by ZFMISC_1:def 2;

thus S1 . p = S1 . (x,y) by A6

.= the addF of F . (x,((comp F) . y)) by A3, A5

.= S2 . (x,y) by A4, A5

.= S2 . p by A6 ; :: thesis: verum