
theorem LM94:
  for A be non empty closed_interval Subset of REAL,
      D be Division of A,
      t be Element of A
    st lower_bound A < D.1 holds
    ex i be Element of NAT st
      i in dom D
    & t in divset(D,i)
    & ( i = 1 or
        (lower_bound divset(D,i) < t
         & t <= (upper_bound divset(D,i)) ) )
proof
  let A be non empty closed_interval Subset of REAL,
      D be Division of A,
      t be Element of A;
  assume AS: lower_bound A < D.1;
  consider i be Element of NAT such that
A24: i in dom D and
A25: t in divset(D,i) by INTEGRA3:3;
  per cases;
  suppose i = 1;
    hence thesis by A24,A25;
  end;
  suppose A26: i <> 1;
    t in [.(lower_bound divset(D,i)),(upper_bound divset(D,i)).]
          by A25,INTEGRA1:4; then
A30: (lower_bound divset(D,i)) <= t
      & t <= (upper_bound divset(D,i)) by XXREAL_1:1;
    thus ex i be Element of NAT st
      i in dom D
    & t in divset(D,i)
    & ( i = 1 or
       (lower_bound divset(D,i) < t
        & t <= (upper_bound divset(D,i)) ) )
    proof
      per cases;
      suppose not lower_bound divset(D,i) = t; then
        lower_bound divset(D,i) < t by A30,XXREAL_0:1;
        hence thesis by A24,A25,A30;
      end;
      suppose A31: lower_bound divset(D,i) = t;
A38:    i-1 in dom D by A24,A26,INTEGRA1:7;
        reconsider j = i- 1 as Element of NAT by A24,A26,INTEGRA1:7;
A40:    t = upper_bound divset(D,j) & t in divset(D,j)
        proof
          j = 1 or j <> 1; then
          upper_bound divset(D,j) = D.(i-1) by A38,INTEGRA1:def 4;
          hence
A41:      t = upper_bound divset(D,j) by A31,A24,A26,INTEGRA1:def 4;
          (lower_bound divset(D,j))
             <= (upper_bound divset(D,j)) by SEQ_4:11; then
          t in [.(lower_bound divset(D,j)),
               (upper_bound divset(D,j)).] by A41;
          hence thesis by INTEGRA1:4;
        end;
        lower_bound divset(D,j) < upper_bound divset(D,j)
        proof
          per cases;
          suppose A42: j = 1; then
            lower_bound divset(D,j)= lower_bound A
            & upper_bound divset(D,j) = D.j by A38,INTEGRA1:def 4;
            hence thesis by AS,A42;
          end;
          suppose A44: j <> 1; then
A45:        ( lower_bound divset(D,j) = D.(j - 1)
              & upper_bound divset(D,j) = D.j) by A38,INTEGRA1:def 4;
            j-1 in dom D by A38,A44,INTEGRA1:7;
            hence thesis by A45,A38,XREAL_1:146,VALUED_0:def 13;
          end;
        end;
        hence thesis by A38,A40;
      end;
    end;
  end;
end;
