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

theorem
  M1 is_similar_to M2 & M2 is Idempotent implies M1 is Idempotent
proof
  assume that
A1: M1 is_similar_to M2 and
A2: M2 is Idempotent;
  consider M4 be Matrix of n,K such that
A3: M4 is invertible and
A4: M1=M4~*M2*M4 by A1;
A5: M4~ is_reverse_of M4 by A3,MATRIX_6:def 4;
A6: width M4=n by MATRIX_0:24;
A7: len M2=n & width M2=n by MATRIX_0:24;
A8: len (M2*M4)=n & width (M4~*M2*M4)=n by MATRIX_0:24;
A9: len (M4~)=n by MATRIX_0:24;
A10: width (M4~*M2)=n by MATRIX_0:24;
A11: width (M4~)=n by MATRIX_0:24;
A12: len M4=n by MATRIX_0:24;
  then M1*M1=(M4~*M2*M4)*(M4~*(M2*M4)) by A4,A7,A11,MATRIX_3:33
    .=((M4~*M2*M4)*M4~)*(M2*M4) by A9,A11,A8,MATRIX_3:33
    .=((M4~*M2)*(M4*M4~))*(M2*M4) by A12,A6,A9,A10,MATRIX_3:33
    .=((M4~*M2)*(1.(K,n)))*(M2*M4) by A5,MATRIX_6:def 2
    .=(M4~*M2)*(M2*M4) by MATRIX_3:19
    .=((M4~*M2)*M2)*M4 by A12,A7,A10,MATRIX_3:33
    .=(M4~*(M2*M2))*M4 by A7,A11,MATRIX_3:33
    .=M1 by A2,A4;
  hence thesis;
end;
