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 Th44:
  for a, b being Complex holds a*(b*z) = (a*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;
  thus (a*b)*z = (multcomplex[;](ab,id COMPLEX))*z by Lm1
    .= multcomplex[;](multcomplex.(aa,bb),id COMPLEX)*z by BINOP_2:def 5
    .= multcomplex[;](aa,multcomplex[;](bb,id COMPLEX))*z by FUNCOP_1:62
    .= (multcomplex[;](aa,id COMPLEX)*multcomplex[;](bb,id COMPLEX))*z by
FUNCOP_1:55
    .= (multcomplex[;](aa,id COMPLEX))*(multcomplex[;](bb,id COMPLEX)*z) by
RELAT_1:36
    .= (multcomplex[;](aa,id COMPLEX))*(b*z) by Lm1
    .= a*(b*z) by Lm1;
end;
