let CNS be ComplexNormSpace; :: thesis: for RNS being RealNormSpace
for h1, h2 being PartFunc of RNS,CNS
for seq being sequence of RNS st rng seq c= (dom h1) /\ (dom h2) holds
( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) )
let RNS be RealNormSpace; :: thesis: for h1, h2 being PartFunc of RNS,CNS
for seq being sequence of RNS st rng seq c= (dom h1) /\ (dom h2) holds
( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) )
let h1, h2 be PartFunc of RNS,CNS; :: thesis: for seq being sequence of RNS st rng seq c= (dom h1) /\ (dom h2) holds
( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) )
let seq be sequence of RNS; :: thesis: ( rng seq c= (dom h1) /\ (dom h2) implies ( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) ) )
A1: (dom h1) /\ (dom h2) c= dom h1 by XBOOLE_1:17;
A2: (dom h1) /\ (dom h2) c= dom h2 by XBOOLE_1:17;
assume A3: rng seq c= (dom h1) /\ (dom h2) ; :: thesis: ( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) )
then A4: rng seq c= dom (h1 + h2) by VFUNCT_1:def 1;

now :: thesis: for n being Nat holds ((h1 + h2) /* seq) . n = ((h1 /* seq) . n) + ((h2 /* seq) . n)

let n be Nat; :: thesis: ((h1 + h2) /* seq) . n = ((h1 /* seq) . n) + ((h2 /* seq) . n)
A5: n in NAT by ORDINAL1:def 12;
A6: seq . n in dom (h1 + h2) by A4, Th6;
thus ((h1 + h2) /* seq) . n = (h1 + h2) /. (seq . n) by A4, FUNCT_2:109, A5
.= (h1 /. (seq . n)) + (h2 /. (seq . n)) by A6, VFUNCT_1:def 1
.= ((h1 /* seq) . n) + (h2 /. (seq . n)) by A3, A1, FUNCT_2:109, XBOOLE_1:1, A5
.= ((h1 /* seq) . n) + ((h2 /* seq) . n) by A3, A2, FUNCT_2:109, XBOOLE_1:1, A5 ; :: thesis: verum

end;

hence (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) by NORMSP_1:def 2; :: thesis: (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq)
A7: rng seq c= dom (h1 - h2) by A3, VFUNCT_1:def 2;

now :: thesis: for n being Nat holds ((h1 - h2) /* seq) . n = ((h1 /* seq) . n) - ((h2 /* seq) . n)

let n be Nat; :: thesis: ((h1 - h2) /* seq) . n = ((h1 /* seq) . n) - ((h2 /* seq) . n)
A8: n in NAT by ORDINAL1:def 12;
A9: seq . n in dom (h1 - h2) by A7, Th6;
thus ((h1 - h2) /* seq) . n = (h1 - h2) /. (seq . n) by A7, FUNCT_2:109, A8
.= (h1 /. (seq . n)) - (h2 /. (seq . n)) by A9, VFUNCT_1:def 2
.= ((h1 /* seq) . n) - (h2 /. (seq . n)) by A3, A1, FUNCT_2:109, XBOOLE_1:1, A8
.= ((h1 /* seq) . n) - ((h2 /* seq) . n) by A3, A2, FUNCT_2:109, XBOOLE_1:1, A8 ; :: thesis: verum

end;

hence (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) by NORMSP_1:def 3; :: thesis: verum