Article JORDAN5A, MML version 4.99.1005

:: JORDAN5A:th 1
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
for b4 being Element of bool the carrier of TOP-REAL b1
      st b4 is_an_arc_of b2,b3
   holds b4 is compact(TOP-REAL b1);

:: JORDAN5A:th 3
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
for b4, b5 being real set
      st ((1 - b4) * b2) + (b4 * b3) = ((1 - b5) * b2) + (b5 * b3) &
         b4 <> b5
   holds b2 = b3;

:: JORDAN5A:th 4
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
      st b2 <> b3
   holds ex b4 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL b1) | LSeg(b2,b3) st
      (for b5 being Element of REAL
             st b5 in [.0,1.]
          holds b4 . b5 = ((1 - b5) * b2) + (b5 * b3)) &
       b4 is being_homeomorphism(I[01], (TOP-REAL b1) | LSeg(b2,b3)) &
       b4 . 0 = b2 &
       b4 . 1 = b3;

:: JORDAN5A:funcreg 1
registration
  let a1 be Element of NAT;
  cluster TOP-REAL a1 -> strict TopSpace-like arcwise_connected;
end;

:: JORDAN5A:exreg 1
registration
  let a1 be Element of NAT;
  cluster non empty strict TopSpace-like compact SubSpace of TOP-REAL a1;
end;

:: JORDAN5A:th 5
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being Path of b1,b2
for b4 being non empty compact SubSpace of TOP-REAL 2
for b5 being Function-like quasi_total Relation of the carrier of I[01],the carrier of b4
      st b3 is one-to-one & b5 = b3 & [#] b4 = rng b3
   holds b5 is being_homeomorphism(I[01], b4);

:: JORDAN5A:th 6
theorem
for b1 being Element of bool REAL holds
      b1 in Family_open_set RealSpace
   iff
      b1 is open;

:: JORDAN5A:th 7
theorem
for b1 being Function-like quasi_total Relation of the carrier of R^1,the carrier of R^1
for b2 being Element of the carrier of R^1
for b3 being Function-like Relation of REAL,REAL
for b4 being Element of REAL
      st b1 is_continuous_at b2 & b1 = b3 & b2 = b4
   holds b3 is_continuous_in b4;

:: JORDAN5A:th 8
theorem
for b1 being Function-like quasi_total continuous Relation of the carrier of R^1,the carrier of R^1
for b2 being Function-like Relation of REAL,REAL
      st b1 = b2
   holds b2 is_continuous_on REAL;

:: JORDAN5A:th 9
theorem
for b1 being Function-like one-to-one quasi_total continuous Relation of the carrier of R^1,the carrier of R^1
   st ex b2, b3 being Element of the carrier of I[01] st
        ex b4, b5, b6, b7 being Element of REAL st
           b2 = b4 & b3 = b5 & b4 < b5 & b6 = b1 . b2 & b7 = b1 . b3 & b7 <= b6
for b2, b3 being Element of the carrier of I[01]
for b4, b5, b6, b7 being Element of REAL
      st b2 = b4 & b3 = b5 & b4 < b5 & b6 = b1 . b2 & b7 = b1 . b3
   holds b7 < b6;

:: JORDAN5A:th 10
theorem
for b1, b2, b3, b4 being Element of REAL
for b5 being Element of the carrier of Closed-Interval-MSpace(b3,b4)
      st b3 <= b4 &
         b5 = b1 &
         0 < b2 &
         ].b1 - b2,b1 + b2.[ c= [.b3,b4.]
   holds ].b1 - b2,b1 + b2.[ = Ball(b5,b2);

:: JORDAN5A:th 11
theorem
for b1, b2 being Element of REAL
for b3 being Element of bool REAL
      st b1 < b2 & not b1 in b3 & not b2 in b3 & b3 in Family_open_set Closed-Interval-MSpace(b1,b2)
   holds b3 is open;

:: JORDAN5A:th 12
theorem
for b1 being open Element of bool REAL
for b2, b3 being Element of REAL
      st b1 c= [.b2,b3.]
   holds not b2 in b1 & not b3 in b1;

:: JORDAN5A:th 13
theorem
for b1, b2 being Element of REAL
for b3 being Element of bool REAL
for b4 being Element of bool the carrier of Closed-Interval-MSpace(b1,b2)
      st b1 <= b2 & b4 = b3 & b3 is open
   holds b4 in Family_open_set Closed-Interval-MSpace(b1,b2);

:: JORDAN5A:th 14
theorem
for b1, b2, b3, b4, b5 being Element of REAL
for b6 being Function-like quasi_total Relation of the carrier of Closed-Interval-TSpace(b1,b2),the carrier of Closed-Interval-TSpace(b3,b4)
for b7 being Element of the carrier of Closed-Interval-TSpace(b1,b2)
for b8 being Function-like Relation of REAL,REAL
      st b1 < b2 & b3 < b4 & b6 is_continuous_at b7 & b6 . b1 = b3 & b6 . b2 = b4 & b6 is one-to-one & b6 = b8 & b7 = b5
   holds b8 | [.b1,b2.] is_continuous_in b5;

:: JORDAN5A:th 15
theorem
for b1, b2, b3, b4 being Element of REAL
for b5 being Function-like quasi_total Relation of the carrier of Closed-Interval-TSpace(b1,b2),the carrier of Closed-Interval-TSpace(b3,b4)
for b6 being Function-like Relation of REAL,REAL
      st b5 is continuous(Closed-Interval-TSpace(b1,b2), Closed-Interval-TSpace(b3,b4)) & b5 is one-to-one & b1 < b2 & b3 < b4 & b5 = b6 & b5 . b1 = b3 & b5 . b2 = b4
   holds b6 is_continuous_on [.b1,b2.];

:: JORDAN5A:th 16
theorem
for b1, b2, b3, b4 being Element of REAL
for b5 being Function-like quasi_total Relation of the carrier of Closed-Interval-TSpace(b1,b2),the carrier of Closed-Interval-TSpace(b3,b4)
   st b1 < b2 & b3 < b4 & b5 is continuous(Closed-Interval-TSpace(b1,b2), Closed-Interval-TSpace(b3,b4)) & b5 is one-to-one & b5 . b1 = b3 & b5 . b2 = b4
for b6, b7 being Element of the carrier of Closed-Interval-TSpace(b1,b2)
for b8, b9, b10, b11 being Element of REAL
      st b6 = b8 & b7 = b9 & b8 < b9 & b10 = b5 . b6 & b11 = b5 . b7
   holds b10 < b11;

:: JORDAN5A:th 17
theorem
for b1 being Function-like one-to-one quasi_total continuous Relation of the carrier of I[01],the carrier of I[01]
   st b1 . 0 = 0 & b1 . 1 = 1
for b2, b3 being Element of the carrier of I[01]
for b4, b5, b6, b7 being Element of REAL
      st b2 = b4 & b3 = b5 & b4 < b5 & b6 = b1 . b2 & b7 = b1 . b3
   holds b6 < b7;

:: JORDAN5A:th 18
theorem
for b1, b2, b3, b4 being Element of REAL
for b5 being Function-like quasi_total Relation of the carrier of Closed-Interval-TSpace(b1,b2),the carrier of Closed-Interval-TSpace(b3,b4)
for b6 being non empty Element of bool the carrier of Closed-Interval-TSpace(b1,b2)
for b7, b8 being Element of bool the carrier of R^1
      st b1 < b2 & b3 < b4 & b7 = b6 & b5 is continuous(Closed-Interval-TSpace(b1,b2), Closed-Interval-TSpace(b3,b4)) & b5 is one-to-one & b7 is compact(R^1) & b5 . b1 = b3 & b5 . b2 = b4 & b5 .: b6 = b8
   holds b5 . lower_bound [#] b7 = lower_bound [#] b8;

:: JORDAN5A:th 19
theorem
for b1, b2, b3, b4 being Element of REAL
for b5 being Function-like quasi_total Relation of the carrier of Closed-Interval-TSpace(b1,b2),the carrier of Closed-Interval-TSpace(b3,b4)
for b6, b7 being non empty Element of bool the carrier of Closed-Interval-TSpace(b1,b2)
for b8, b9 being Element of bool the carrier of R^1
      st b1 < b2 & b3 < b4 & b8 = b6 & b9 = b7 & b5 is continuous(Closed-Interval-TSpace(b1,b2), Closed-Interval-TSpace(b3,b4)) & b5 is one-to-one & b8 is compact(R^1) & b5 . b1 = b3 & b5 . b2 = b4 & b5 .: b6 = b7
   holds b5 . upper_bound [#] b8 = upper_bound [#] b9;

:: JORDAN5A:th 20
theorem
for b1, b2 being real set
      st b1 <= b2
   holds lower_bound [.b1,b2.] = b1 & upper_bound [.b1,b2.] = b2;

:: JORDAN5A:th 21
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being Element of REAL
for b9 being Function-like quasi_total Relation of the carrier of Closed-Interval-TSpace(b1,b2),the carrier of Closed-Interval-TSpace(b3,b4)
      st b1 < b2 & b3 < b4 & b5 < b6 & b1 <= b5 & b6 <= b2 & b9 is being_homeomorphism(Closed-Interval-TSpace(b1,b2), Closed-Interval-TSpace(b3,b4)) & b9 . b1 = b3 & b9 . b2 = b4 & b7 = b9 . b5 & b8 = b9 . b6
   holds b9 .: [.b5,b6.] = [.b7,b8.];

:: JORDAN5A:th 22
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3, b4 being Element of the carrier of TOP-REAL 2
      st b1 meets b2 & b1 /\ b2 is closed(TOP-REAL 2) & b1 is_an_arc_of b3,b4
   holds ex b5 being Element of the carrier of TOP-REAL 2 st
      b5 in b1 /\ b2 &
       (ex b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | b1 st
          ex b7 being Element of REAL st
             b6 is being_homeomorphism(I[01], (TOP-REAL 2) | b1) &
              b6 . 0 = b3 &
              b6 . 1 = b4 &
              b6 . b7 = b5 &
              0 <= b7 &
              b7 <= 1 &
              (for b8 being Element of REAL
                    st 0 <= b8 & b8 < b7
                 holds not b6 . b8 in b2));

:: JORDAN5A:th 23
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3, b4 being Element of the carrier of TOP-REAL 2
      st b1 meets b2 & b1 /\ b2 is closed(TOP-REAL 2) & b1 is_an_arc_of b3,b4
   holds ex b5 being Element of the carrier of TOP-REAL 2 st
      b5 in b1 /\ b2 &
       (ex b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | b1 st
          ex b7 being Element of REAL st
             b6 is being_homeomorphism(I[01], (TOP-REAL 2) | b1) &
              b6 . 0 = b3 &
              b6 . 1 = b4 &
              b6 . b7 = b5 &
              0 <= b7 &
              b7 <= 1 &
              (for b8 being Element of REAL
                    st b8 <= 1 & b7 < b8
                 holds not b6 . b8 in b2));