
theorem
  for M being Matrix of COMPLEX holds M+M+M = 3*M
proof
  reconsider rr=1+1 as Element of COMPLEX by XCMPLX_0:def 2;
  reconsider e3=(1_F_Complex) + (1_F_Complex)+ (1_F_Complex) as Element of
  F_Complex;
  let M be Matrix of COMPLEX;
A1: len M =len COMPLEX2Field M & width M=width COMPLEX2Field M;
A2: (1_F_Complex) + (1_F_Complex) =(the addF of F_Complex).(1_F_Complex,
  1_F_Complex) by RLVECT_1:2
    .=addcomplex.(1r,1r) by COMPLFLD:8,def 1
    .=1+1 by BINOP_2:def 3,COMPLEX1:def 4;
  (1_F_Complex) + (1_F_Complex) + (1_F_Complex) = (the addF of F_Complex).
  (((1_F_Complex) + (1_F_Complex)),1_F_Complex) by RLVECT_1:2
    .= addcomplex.(1+1,1) by A2,COMPLFLD:def 1
    .= rr + 1 by BINOP_2:def 3
    .= 3;
  then 3*M= Field2COMPLEX (e3*(COMPLEX2Field M)) by Def7
    .=Field2COMPLEX ((1_F_Complex)*(COMPLEX2Field M) + ((1_F_Complex)+ (
  1_F_Complex))*(COMPLEX2Field M)) by Th12
    .=Field2COMPLEX (COMPLEX2Field M + ((1_F_Complex)+ (1_F_Complex))*(
  COMPLEX2Field M)) by Th9
    .=Field2COMPLEX (COMPLEX2Field M + ((1_F_Complex)*(COMPLEX2Field M)+ (
  1_F_Complex)*(COMPLEX2Field M))) by Th12
    .=Field2COMPLEX (COMPLEX2Field M + (COMPLEX2Field M + (1_F_Complex)*(
  COMPLEX2Field M))) by Th9
    .=Field2COMPLEX (COMPLEX2Field M+(COMPLEX2Field M+COMPLEX2Field M)) by Th9
    .=Field2COMPLEX (COMPLEX2Field M+COMPLEX2Field M+COMPLEX2Field M) by A1,
MATRIX_3:3;
  hence thesis;
end;
