reserve x,y for set,
  i for Nat;
reserve V for non empty CLSStruct,
  u,v,v1,v2,v3 for VECTOR of V,
  A for Subset of V,
  l, l1, l2 for C_Linear_Combination of A,
  x,y,y1,y2 for set,
  a,b for Complex,
  F for FinSequence of the carrier of V,
  f for Function of the carrier of V, COMPLEX;
reserve K,L,L1,L2,L3 for C_Linear_Combination of V;
reserve e,e1,e2 for Element of C_LinComb V;

theorem Th42:
  for a being Real, z being Complex holds Re(a*z)=a* Re(z)
proof
  let a be Real;
  let z be Complex;
  Re(a * z) = Re a * Re z - Im a * Im z by COMPLEX1:9
    .= Re a * Re z - 0 * Im z by COMPLEX1:def 2
    .= a * Re (z) by COMPLEX1:def 1;
  hence thesis;
end;
