let it1, it2 be BinOp of (Funcs (X, the carrier of Y)); :: thesis: ( ( for f, g being Element of Funcs (X, the carrier of Y) holds it1 . (f,g) = the addF of Y .: (f,g) ) & ( for f, g being Element of Funcs (X, the carrier of Y) holds it2 . (f,g) = the addF of Y .: (f,g) ) implies it1 = it2 )
assume that
A1: for f, g being Element of Funcs (X, the carrier of Y) holds it1 . (f,g) = the addF of Y .: (f,g) and
A2: for f, g being Element of Funcs (X, the carrier of Y) holds it2 . (f,g) = the addF of Y .: (f,g) ; :: thesis: it1 = it2
now :: thesis: for f, g being Element of Funcs (X, the carrier of Y) holds it1 . (f,g) = it2 . (f,g)
let f, g be Element of Funcs (X, the carrier of Y); :: thesis: it1 . (f,g) = it2 . (f,g)
thus it1 . (f,g) = the addF of Y .: (f,g) by A1
.= it2 . (f,g) by A2 ; :: thesis: verum
end;
hence it1 = it2 ; :: thesis: verum