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

theorem
  n>0 implies (M1 is Idempotent iff M1@ is Idempotent)
proof
  assume
A1: n>0;
  set M2=M1@;
A2: width M1=n & len M1=n by MATRIX_0:24;
  thus M1 is Idempotent implies M2 is Idempotent
  by A1,A2,MATRIX_3:22;
A3: width M2=n & len M2=n by MATRIX_0:24;
  assume
A4: M2 is Idempotent;
  M1=M2@ by A1,A2,MATRIX_0:57
    .=(M2*M2)@ by A4
    .=M2@*M2@ by A1,A3,MATRIX_3:22
    .=M1*(M1@)@ by A1,A2,MATRIX_0:57
    .=M1*M1 by A1,A2,MATRIX_0:57;
  hence thesis;
end;
