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;
reserve h for Function of COMPLEX,COMPLEX,
  g for Function of REAL,REAL;

theorem Th53:
  for a, b being Complex holds (a-b)*z = a*z - b*z
proof
  let a, b be Complex;
  reconsider aa = a, bb = b as Element of COMPLEX by XCMPLX_0:def 2;
  (a - b)*z=(a+(-b))*z .= aa*z +(-bb)*z by Th52
    .= a*z - b*z by Th45;
  hence thesis;
end;
