reserve i,n for Nat,
  K for Field,
  M1,M2,M3,M4 for Matrix of n,K;

theorem
  M2*M1=1.(K,n) implies M1*M2 is Idempotent
proof
  assume
A1: M2*M1=1.(K,n);
A2: len M1=n & width M1=n by MATRIX_0:24;
A3: width M2=n by MATRIX_0:24;
A4: len M2=n by MATRIX_0:24;
  width (M1*M2)=n by MATRIX_0:24;
  then M1*M2*(M1*M2)=(M1*M2*M1)*M2 by A2,A4,MATRIX_3:33
    .=(M1*(1.(K,n)))*M2 by A1,A2,A4,A3,MATRIX_3:33
    .=M1*M2 by MATRIX_3:19;
  hence thesis;
end;
