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 Th52:
  for a, b being Complex holds (a+b)*z = a*z + b*z
proof
  let a, b be Complex;
  reconsider aa = a, bb = b, ab=a+b as Element of COMPLEX by XCMPLX_0:def 2;
  set c1M = multcomplex[;](aa,id COMPLEX), c2M = multcomplex[;](bb,id COMPLEX);
  thus (a + b)*z = (multcomplex[;](ab,id COMPLEX))*z by Lm1
    .= multcomplex[;](addcomplex.(aa,bb),id COMPLEX)*z by BINOP_2:def 3
    .= addcomplex.:(c1M,c2M)*z by FINSEQOP:35,SEQ_4:54
    .= addcomplex.:(c1M*z,c2M*z) by FUNCOP_1:25
    .= c1M*z + c2M*z by SEQ_4:def 6
    .= a*z + c2M*z by Lm1
    .= a*z + b*z by Lm1;
end;
