reserve x for set,
  D for non empty set,
  k,n,m,i,j,l for Nat,
  K for Field;

theorem
  for A,B being Matrix of n,REAL holds 
  len (A*B)=len A & width (A*B)=width B & 
  len (A*B)=n & width (A*B)=n
proof
  let A,B be Matrix of n,REAL;
A1: len B=n by MATRIX_0:25;
A2: len A=n by MATRIX_0:25;
  per cases;
  suppose
A3: n>0;
    then width MXR2MXF A =n by MATRIX_0:23
      .=len MXR2MXF B by MATRIX_0:25;
    then len (A*B)=len A & width (A*B)=width B by MATRIX_3:def 4;
    hence thesis by A1,A3,MATRIX_0:20,25;
  end;
  suppose
A4: n<=0;
    then
A5: width (MXR2MXF A) =0 by A2,MATRIX_0:def 3
      .=len (MXR2MXF B) by A4,MATRIX_0:25;
    then len (A*B) = len A by MATRIX_3:def 4;
    hence thesis by A2,A4,A5,MATRIX_0:def 3,MATRIX_3:def 4;
  end;
end;
