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+M1+M1 is_similar_to M2+M2+M2
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: len M4=n & len (M4~*M2)=n by MATRIX_0:24;
A6: width (M4~*M2)=n by MATRIX_0:24;
A7: len M2=n & width M2=n by MATRIX_0:24;
A8: len (M4~)=n & width (M4~)=n by MATRIX_0:24;
  then
A9: M4~*(M2+M2)*M4=(M4~*M2+M4~*M2)*M4 by A7,MATRIX_4:62
    .=M1+M1 by A4,A5,A6,MATRIX_4:63;
  take M4;
A10: len (M4~*(M2+M2))=n & width (M4~*(M2+M2))=n by MATRIX_0:24;
  len (M2+M2)=n & width (M2+M2)=n by MATRIX_0:24;
  then M4~*(M2+M2+M2)*M4=(M4~*(M2+M2)+M4~*M2)*M4 by A7,A8,MATRIX_4:62
    .=M1+M1+M1 by A4,A5,A6,A10,A9,MATRIX_4:63;
  hence thesis by A3;
end;
