reserve i,j,n,k for Nat,
  a for Element of COMPLEX,
  R1,R2 for Element of i-tuples_on COMPLEX;

theorem Th2:
  for a being Complex,M being Matrix of COMPLEX holds len (a
  *M)=len M & width (a*M)=width M
proof
  let a be Complex,M be Matrix of COMPLEX;
  a in COMPLEX by XCMPLX_0:def 2;
  then reconsider ea=a as Element of F_Complex by COMPLFLD:def 1;
A1: width (a*M)=width COMPLEX2Field (a*M) by MATRIX_5:7
    .= width COMPLEX2Field Field2COMPLEX (ea*(COMPLEX2Field M)) by
MATRIX_5:def 7
    .= width (ea*(COMPLEX2Field M)) by MATRIX_5:6
    .= width COMPLEX2Field M by MATRIX_3:def 5
    .= width M by MATRIX_5:def 1;
  len (a*M)=len COMPLEX2Field (a*M) by MATRIX_5:7
    .= len COMPLEX2Field Field2COMPLEX (ea*(COMPLEX2Field M)) by MATRIX_5:def 7
    .= len (ea*(COMPLEX2Field M)) by MATRIX_5:6
    .= len COMPLEX2Field M by MATRIX_3:def 5
    .= len M by MATRIX_5:def 1;
  hence thesis by A1;
end;
