theorem
  (c1 + c2)*z = c1*z + c2*z
proof
  set c1M = multcomplex[;](c1,id COMPLEX), c2M = multcomplex[;](c2,id COMPLEX);
  thus (c1 + c2)*z = multcomplex[;](addcomplex.(c1,c2),id COMPLEX)*z by
BINOP_2:def 3
    .= addcomplex.:(c1M,c2M)*z by Th54,FINSEQOP:35
    .= c1*z + c2*z by FUNCOP_1:25;
end;
