let X, Y be non empty set ; :: thesis: for V being RealNormSpace
for g, f being PartFunc of X, the carrier of V
for g1, f1 being PartFunc of Y, the carrier of V st g = g1 & f = f1 holds
g1 - f1 = g - f
let V be RealNormSpace; :: thesis: for g, f being PartFunc of X, the carrier of V
for g1, f1 being PartFunc of Y, the carrier of V st g = g1 & f = f1 holds
g1 - f1 = g - f
let g, f be PartFunc of X, the carrier of V; :: thesis: for g1, f1 being PartFunc of Y, the carrier of V st g = g1 & f = f1 holds
g1 - f1 = g - f
let g1, f1 be PartFunc of Y, the carrier of V; :: thesis: ( g = g1 & f = f1 implies g1 - f1 = g - f )
assume A1: ( g = g1 & f = f1 ) ; :: thesis: g1 - f1 = g - f
A2: dom (g - f) = (dom g) /\ (dom f) by VFUNCT_1:def 2
.= (dom g1) /\ (dom f1) by A1 ;
A3: g - f is PartFunc of Y, the carrier of V by A2, RELSET_1:5;
for c being Element of Y st c in dom (g - f) holds
(g - f) /. c = (g1 /. c) - (f1 /. c) by A1, VFUNCT_1:def 2;
hence g1 - f1 = g - f by A3, A2, VFUNCT_1:def 2; :: thesis: verum