let X be non empty compact TopSpace; :: thesis: for f, g, h being RealMap of X
for F, G, H being Point of st f = F & g = G & h = H holds
( H = F - G iff for x being Element of holds h . x = (f . x) - (g . x) )
let f, g, h be RealMap of X; :: thesis: for F, G, H being Point of st f = F & g = G & h = H holds
( H = F - G iff for x being Element of holds h . x = (f . x) - (g . x) )
let F, G, H be Point of ; :: thesis: ( f = F & g = G & h = H implies ( H = F - G iff for x being Element of holds h . x = (f . x) - (g . x) ) )
assume A1: ( f = F & g = G & h = H ) ; :: thesis: ( H = F - G iff for x being Element of holds h . x = (f . x) - (g . x) )

A2: now

assume H = F - G ; :: thesis: for x being Element of holds h . x = (f . x) - (g . x)
then H + G = F - (G - G) by RLVECT_1:43;
then H + G = F - (0. (R_Normed_Algebra_of_ContinuousFunctions X)) by RLVECT_1:28;
then A3: H + G = F by RLVECT_1:26;

now

let x be Element of ; :: thesis: (f . x) - (g . x) = h . x
f . x = (h . x) + (g . x) by A1, A3, ThB22;
hence (f . x) - (g . x) = h . x ; :: thesis: verum

end;

hence for x being Element of holds h . x = (f . x) - (g . x) ; :: thesis: verum

end;

now

assume A4: for x being Element of holds h . x = (f . x) - (g . x) ; :: thesis: F - G = H

now

let x be Element of ; :: thesis: (h . x) + (g . x) = f . x
h . x = (f . x) - (g . x) by A4;
hence (h . x) + (g . x) = f . x ; :: thesis: verum

end;

then F = H + G by A1, ThB22;
then F - G = H + (G - G) by RLVECT_1:def 6;
then F - G = H + (0. (R_Normed_Algebra_of_ContinuousFunctions X)) by RLVECT_1:28;
hence F - G = H by RLVECT_1:10; :: thesis: verum

end;

hence ( H = F - G iff for x being Element of holds h . x = (f . x) - (g . x) ) by A2; :: thesis: verum