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

A2: now :: thesis: ( H = F - G implies for x being Element of X holds h . x = (f . x) - (g . x) )

assume H = F - G ; :: thesis: for x being Element of X holds h . x = (f . x) - (g . x)
then H + G = F - (G - G) by RLVECT_1:29;
then H + G = F - (0. (R_Normed_Algebra_of_ContinuousFunctions X)) by RLVECT_1:15;
then A3: H + G = F by RLVECT_1:13;

now :: thesis: for x being Element of X holds (f . x) - (g . x) = h . x

let x be Element of X; :: thesis: (f . x) - (g . x) = h . x
f . x = (h . x) + (g . x) by A1, A3, Th15;
hence (f . x) - (g . x) = h . x ; :: thesis: verum

end;

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

end;

now :: thesis: ( ( for x being Element of X holds h . x = (f . x) - (g . x) ) implies F - G = H )

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

now :: thesis: for x being Element of X holds (h . x) + (g . x) = f . x

let x be Element of X; :: 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, Th15;
then F - G = H + (G - G) by RLVECT_1:def 3;
then F - G = H + (0. (R_Normed_Algebra_of_ContinuousFunctions X)) by RLVECT_1:15;
hence F - G = H by RLVECT_1:4; :: thesis: verum

end;

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