reserve i,j for Element of NAT,
  x,y,z for FinSequence of COMPLEX,
  c for Element of COMPLEX,
  R,R1,R2 for Element of i-tuples_on COMPLEX;
reserve C for Function of [:COMPLEX,COMPLEX:],COMPLEX;
reserve G for Function of [:REAL,REAL:],REAL;

theorem Th40:
  for x being FinSequence of COMPLEX holds len (Re x)=len x & len (Im x)=len x
proof
  let x be FinSequence of COMPLEX;
A1: len x=len (x*') by Def1;
A2: len (Im x) =len (x-x*') by Th3
    .=len x by A1,Th7;
  len (Re x) =len (x+x*') by Th3
    .=len x by A1,Th6;
  hence thesis by A2;
end;
