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

theorem
  M1 is_similar_to M2 implies M1+1.(K,n) is_similar_to M2+1.(K,n)
proof
  assume M1 is_similar_to M2; then
  consider M4 be Matrix of n,K such that
A3: M4 is invertible and
A4: M1=M4~*M2*M4;
A5: M4~ is_reverse_of M4 by A3,MATRIX_6:def 4;
A6: len (1.(K,n))=n & width (1.(K,n))=n by MATRIX_0:24;
A7: width (M4~*M2)=n by MATRIX_0:24;
A8: len M4=n & len (M4~*M2)=n by MATRIX_0:24;
  take M4;
A9: len (M4~)=n & width (M4~)=n by MATRIX_0:24;
  len M2=n & width M2=n by MATRIX_0:24;
  then M4~*(M2+1.(K,n))*M4=(M4~*M2+M4~*(1.(K,n)))*M4 by A9,A6,MATRIX_4:62
    .=(M4~*M2+M4~)*M4 by MATRIX_3:19
    .=M4~*M2*M4 +M4~*M4 by A9,A8,A7,MATRIX_4:63
    .=M1+1.(K,n) by A4,A5,MATRIX_6:def 2;
  hence thesis by A3;
end;
