let X, Y, Z be set ; for f1, f2 being Function of X,Y
for g1, g2 being Function of X,Z st ( Y = {} implies X = {} ) & ( Z = {} implies X = {} ) & <:f1,g1:> = <:f2,g2:> holds
( f1 = f2 & g1 = g2 )
let f1, f2 be Function of X,Y; for g1, g2 being Function of X,Z st ( Y = {} implies X = {} ) & ( Z = {} implies X = {} ) & <:f1,g1:> = <:f2,g2:> holds
( f1 = f2 & g1 = g2 )
let g1, g2 be Function of X,Z; ( ( Y = {} implies X = {} ) & ( Z = {} implies X = {} ) & <:f1,g1:> = <:f2,g2:> implies ( f1 = f2 & g1 = g2 ) )
assume that
A1:
( Y = {} implies X = {} )
and
A2:
( Z = {} implies X = {} )
; ( not <:f1,g1:> = <:f2,g2:> or ( f1 = f2 & g1 = g2 ) )
A3:
( dom g1 = X & dom g2 = X )
by A2, FUNCT_2:def 1;
( dom f1 = X & dom f2 = X )
by A1, FUNCT_2:def 1;
hence
( not <:f1,g1:> = <:f2,g2:> or ( f1 = f2 & g1 = g2 ) )
by A3, Th54; verum