let seq be Complex_Sequence; :: thesis: for h being PartFunc of COMPLEX,COMPLEX st rng seq c= dom (h ^) holds
h /* seq is non-zero
let h be PartFunc of COMPLEX,COMPLEX; :: thesis: ( rng seq c= dom (h ^) implies h /* seq is non-zero )
assume A1: rng seq c= dom (h ^) ; :: thesis: h /* seq is non-zero
then A2: ( (dom h) \ (h " {0c}) c= dom h & rng seq c= (dom h) \ (h " {0c}) ) by CFUNCT_1:def 2, XBOOLE_1:36;
now :: thesis: for n being Element of NAT holds not (h /* seq) . n = 0c
given n being Element of NAT such that A3: (h /* seq) . n = 0c ; :: thesis: contradiction
seq . n in rng seq by VALUED_0:28;
then seq . n in dom (h ^) by A1;
then seq . n in (dom h) \ (h " {0c}) by CFUNCT_1:def 2;
then ( seq . n in dom h & not seq . n in h " {0c} ) by XBOOLE_0:def 5;
then A4: not h /. (seq . n) in {0c} by PARTFUN2:26;
h /. (seq . n) = 0c by A2, A3, FUNCT_2:109, XBOOLE_1:1;
hence contradiction by A4, TARSKI:def 1; :: thesis: verum
end;
hence h /* seq is non-zero by COMSEQ_1:4; :: thesis: verum