reserve i,j for Nat;

theorem Th40:
  for a being Real,A,B being Matrix of REAL st width A=len B holds
  A*(a*B)=a*(A*B)
proof
  let a be Real,A,B be Matrix of REAL;
  assume
A1: width A=len B;
  reconsider ea=a as Element of F_Real by XREAL_0:def 1;
  thus A*(a*B)=MXF2MXR((MXR2MXF A)*(MXR2MXF(MXF2MXR(ea*(MXR2MXF B))))) by Def7
    .=MXF2MXR(ea*(MXR2MXF (MXF2MXR((MXR2MXF A)*(MXR2MXF B))))) by A1,Th22
    .=a*(A*B) by Def7;
end;
