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 Nilpotent implies M1 is Nilpotent
proof
  assume that
A1: M1 is_similar_to M2 and
A2: M2 is Nilpotent;
  consider M4 be Matrix of n,K such that
A4: M4 is invertible and
A5: M1=M4~*M2*M4 by A1;
A6: M4~ is_reverse_of M4 by A4,MATRIX_6:def 4;
A7: width M4=n by MATRIX_0:24;
A8: width (M4~)=n by MATRIX_0:24;
A9: width (M4~*M2)=n by MATRIX_0:24;
A10: len (M4~)=n by MATRIX_0:24;
A11: len (M2*M4)=n & width (M4~*M2*M4)=n by MATRIX_0:24;
A12: len M4=n by MATRIX_0:24;
A13: len M2=n & width M2=n by MATRIX_0:24;
  then M1*M1=(M4~*M2*M4)*(M4~*(M2*M4)) by A5,A12,A8,MATRIX_3:33
    .=((M4~*M2*M4)*M4~)*(M2*M4) by A10,A8,A11,MATRIX_3:33
    .=((M4~*M2)*(M4*M4~))*(M2*M4) by A12,A7,A10,A9,MATRIX_3:33
    .=((M4~*M2)*(1.(K,n)))*(M2*M4) by A6,MATRIX_6:def 2
    .=(M4~*M2)*(M2*M4) by MATRIX_3:19
    .=((M4~*M2)*M2)*M4 by A12,A13,A9,MATRIX_3:33
    .=(M4~*(M2*M2))*M4 by A13,A8,MATRIX_3:33
    .=M4~*(0.(K,n))*M4 by A2
    .=(0.(K,n,n))*M4 by A10,A8,MATRIX_6:2
    .=0.(K,n) by A12,A7,MATRIX_6:1;
  hence thesis;
end;
