reserve i,j for Nat;

theorem
  for a being Real, A,B being Matrix of REAL st len A=len B &
  width A=width B holds a*(A + B) = a*A + a*B
proof
  let a be Real, A,B be Matrix of REAL;
  assume
A1: len A=len B & width A=width B;
  reconsider ea=a as Element of F_Real by XREAL_0:def 1;
A2: a*A + a*B = MXF2MXR ((MXR2MXF (MXF2MXR (ea*(MXR2MXF A)))+(MXR2MXF (a*B))
  )) by Def7
    .= MXF2MXR (((ea*(MXR2MXF A))+(MXR2MXF (MXF2MXR (ea*(MXR2MXF B)))))) by
Def7
    .= MXF2MXR ((ea*(MXR2MXF A)+(ea*(MXR2MXF B))));
  a*(A + B) = MXF2MXR (ea*(MXR2MXF (A+B))) by Def7
    .= MXF2MXR (ea*(MXR2MXF A)+ea*(MXR2MXF B)) by A1,MATRIX_5:20;
  hence thesis by A2;
end;
