reserve i,j,m,n,k for Nat,
  x,y for set,
  K for Field,
  a,L for Element of K;

theorem Th9:
  Jordan_block(L,n) + a*1.(K,n)=Jordan_block(L+a,n)
proof
  set J=Jordan_block(L,n);
  set Ja=Jordan_block(L+a,n);
  set ONE=1.(K,n);
  now
A1: Indices J=Indices Ja by MATRIX_0:26;
    let i,j such that
A2: [i,j] in Indices Ja;
A3: Indices J=Indices ONE by MATRIX_0:26;
    now
      per cases;
      suppose
A4:     i=j;
        hence Ja*(i,j) = L+a by A2,Def1
          .= J*(i,j)+a by A2,A1,A4,Def1
          .= J*(i,j)+a*1_K
          .= J*(i,j)+a*(ONE*(i,j)) by A2,A1,A3,A4,MATRIX_1:def 3
          .= J*(i,j)+(a*ONE)*(i,j) by A2,A1,A3,MATRIX_3:def 5
          .= (J+a*ONE)*(i,j) by A2,A1,MATRIX_3:def 3;
      end;
      suppose
A5:     i+1=j;
        then
A6:     i<>j;
        thus Ja*(i,j) = 1_K by A2,A5,Def1
          .= 1_K+0.K by RLVECT_1:def 4
          .= J*(i,j)+0.K by A2,A1,A5,Def1
          .= J*(i,j)+a*0.K
          .= J*(i,j)+a*(ONE*(i,j)) by A2,A1,A3,A6,MATRIX_1:def 3
          .= J*(i,j)+(a*ONE)*(i,j) by A2,A1,A3,MATRIX_3:def 5
          .= (J+a*ONE)*(i,j) by A2,A1,MATRIX_3:def 3;
      end;
      suppose
A7:     i<>j & i+1<>j;
        hence Ja*(i,j) = 0.K by A2,Def1
          .= 0.K+0.K by RLVECT_1:def 4
          .= J*(i,j)+0.K by A2,A1,A7,Def1
          .= J*(i,j)+a*0.K
          .= J*(i,j)+a*(ONE*(i,j)) by A2,A1,A3,A7,MATRIX_1:def 3
          .= J*(i,j)+(a*ONE)*(i,j) by A2,A1,A3,MATRIX_3:def 5
          .= (J+a*ONE)*(i,j) by A2,A1,MATRIX_3:def 3;
      end;
    end;
    hence Ja*(i,j)=(J+a*ONE)*(i,j);
  end;
  hence thesis by MATRIX_0:27;
end;
