let seq be Complex_Sequence; :: thesis: for h1, h2 being PartFunc of COMPLEX,COMPLEX st rng seq c= (dom h1) /\ (dom h2) holds
( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) & (h1 (#) h2) /* seq = (h1 /* seq) (#) (h2 /* seq) )
let h1, h2 be PartFunc of COMPLEX,COMPLEX; :: thesis: ( rng seq c= (dom h1) /\ (dom h2) implies ( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (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) & (h1 (#) h2) /* seq = (h1 /* seq) (#) (h2 /* seq) )
then A4: rng seq c= dom (h1 + h2) by VALUED_1:def 1;

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

let n be Element of NAT ; :: thesis: ((h1 + h2) /* seq) . n = ((h1 /* seq) + (h2 /* seq)) . n
A5: seq . n in rng seq by VALUED_0:28;
thus ((h1 + h2) /* seq) . n = (h1 + h2) /. (seq . n) by A4, FUNCT_2:109
.= (h1 /. (seq . n)) + (h2 /. (seq . n)) by A4, A5, CFUNCT_1:1
.= ((h1 /* seq) . n) + (h2 /. (seq . n)) by A3, A1, FUNCT_2:109, XBOOLE_1:1
.= ((h1 /* seq) . n) + ((h2 /* seq) . n) by A3, A2, FUNCT_2:109, XBOOLE_1:1
.= ((h1 /* seq) + (h2 /* seq)) . n by VALUED_1:1 ; :: thesis: verum

end;

hence (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) by FUNCT_2:63; :: thesis: ( (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) & (h1 (#) h2) /* seq = (h1 /* seq) (#) (h2 /* seq) )
A6: rng seq c= dom (h1 - h2) by A3, CFUNCT_1:2;

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)
A7: n in NAT by ORDINAL1:def 12;
A8: seq . n in rng seq by VALUED_0:28;
thus ((h1 - h2) /* seq) . n = (h1 - h2) /. (seq . n) by A6, FUNCT_2:109, A7
.= (h1 /. (seq . n)) - (h2 /. (seq . n)) by A6, A8, CFUNCT_1:2
.= ((h1 /* seq) . n) - (h2 /. (seq . n)) by A3, A1, FUNCT_2:109, XBOOLE_1:1, A7
.= ((h1 /* seq) . n) - ((h2 /* seq) . n) by A3, A2, FUNCT_2:109, XBOOLE_1:1, A7 ; :: thesis: verum

end;

hence (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) by Th1; :: thesis: (h1 (#) h2) /* seq = (h1 /* seq) (#) (h2 /* seq)
A9: rng seq c= dom (h1 (#) h2) by A3, VALUED_1:def 4;

now :: thesis: for n being Element of NAT holds ((h1 (#) h2) /* seq) . n = ((h1 /* seq) (#) (h2 /* seq)) . n

let n be Element of NAT ; :: thesis: ((h1 (#) h2) /* seq) . n = ((h1 /* seq) (#) (h2 /* seq)) . n
A10: seq . n in rng seq by VALUED_0:28;
thus ((h1 (#) h2) /* seq) . n = (h1 (#) h2) /. (seq . n) by A9, FUNCT_2:109
.= (h1 /. (seq . n)) * (h2 /. (seq . n)) by A9, A10, CFUNCT_1:3
.= ((h1 /* seq) . n) * (h2 /. (seq . n)) by A3, A1, FUNCT_2:109, XBOOLE_1:1
.= ((h1 /* seq) . n) * ((h2 /* seq) . n) by A3, A2, FUNCT_2:109, XBOOLE_1:1
.= ((h1 /* seq) (#) (h2 /* seq)) . n by VALUED_1:5 ; :: thesis: verum

end;

hence (h1 (#) h2) /* seq = (h1 /* seq) (#) (h2 /* seq) by FUNCT_2:63; :: thesis: verum