reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  f,g for PartFunc of X,COMPLEX,
  r for Real,
  k for Real,
  n for Nat,
  E for Element of S;
reserve x,A for set;
reserve c for FinSequence of COMPLEX;

theorem Th46:
  for n be Nat st n in dom Re c holds (Re c).n = Re(c.n)
proof
  let n be Nat;
A1: (1/2*c*').n = 1/2*(c*'.n) by COMPLSP2:16;
  len(1/2*c) = len c & len(1/2*(c*')) = len(c*') by COMPLSP2:3;
  then
A2: len(1/2*c) = len(1/2*(c*')) by COMPLSP2:def 1;
  len(c*') = len c by COMPLSP2:def 1;
  then n in NAT & 1/2*(c + c*') = 1/2*c + 1/2*c*' by COMPLSP2:30
,ORDINAL1:def 12;
  then
A3: (Re c).n = (1/2*c).n + (1/2*c*').n by A2,COMPLSP2:26;
  assume
A4: n in dom Re c;
  then n <= len Re c by FINSEQ_3:25;
  then
A5: n <= len c by COMPLSP2:48;
  1 <= n by A4,FINSEQ_3:25;
  then (1/2*c*').n = 1/2*(c.n)*' by A5,A1,COMPLSP2:def 1;
  then
A6: (Re c).n = 1/2*(c.n) + 1/2*(c.n)*' by A3,COMPLSP2:16;
  c.n = Re(c.n) + (Im(c.n))*<i> by COMPLEX1:13;
  hence thesis by A6;
end;
