let X be non empty set ; :: thesis: for Y being set
for F being BinOp of X
for g being Function of Y,X
for x being Element of X holds F [;] x,g is Function of Y,X
let Y be set ; :: thesis: for F being BinOp of X
for g being Function of Y,X
for x being Element of X holds F [;] x,g is Function of Y,X
let F be BinOp of X; :: thesis: for g being Function of Y,X
for x being Element of X holds F [;] x,g is Function of Y,X
let g be Function of Y,X; :: thesis: for x being Element of X holds F [;] x,g is Function of Y,X
let x be Element of X; :: thesis: F [;] x,g is Function of Y,X
dom g = Y by FUNCT_2:def 1;
then reconsider f = (dom g) --> x as Function of Y,X ;
F * <:f,g:> is Function of Y,X ;
hence F [;] x,g is Function of Y,X ; :: thesis: verum