reserve i,j for Nat;

theorem
  for A being Matrix of REAL holds A+A+A=3*A
proof
  reconsider e3=(1_F_Real) + (1_F_Real) + (1_F_Real) as Element of F_Real;
  let A be Matrix of REAL;
A1: len A =len MXR2MXF A & width A=width MXR2MXF A;
  3*A= MXF2MXR (e3*(MXR2MXF A)) by Def7
    .=MXF2MXR ((1_F_Real)*(MXR2MXF A) + ((1_F_Real)+ (1_F_Real))*(MXR2MXF A)
  ) by MATRIX_5:12
    .=MXF2MXR ((MXR2MXF A) + ((1_F_Real)+ (1_F_Real))*(MXR2MXF A)) by
MATRIX_5:9
    .=MXF2MXR (MXR2MXF A + ((1_F_Real)*(MXR2MXF A)+ (1_F_Real)*(MXR2MXF A)))
  by MATRIX_5:12
    .=MXF2MXR (MXR2MXF A + (MXR2MXF A + (1_F_Real)*(MXR2MXF A))) by MATRIX_5:9
    .=MXF2MXR (MXR2MXF A+(MXR2MXF A+MXR2MXF A)) by MATRIX_5:9
    .=MXF2MXR (MXR2MXF A+MXR2MXF A+MXR2MXF A) by A1,MATRIX_3:3;
  hence thesis;
end;
