 reserve n for Nat;
 reserve s1 for sequence of Euclid n,
         s2 for sequence of REAL-NS n;
reserve r,s for Real;

theorem
  for a,b,c being Real,Iab,Iac,Icb being non empty compact Subset of REAL st
  a <= c <= b & Iab = [.a,b.] & Iac=[.a,c.] & Icb=[.c,b.] holds
  for Dac being Division of Iac, Dcb being Division of Icb st c < Dcb.1 holds
  Dac ^ Dcb is Division of Iab
  proof
    let a,b,c be Real,
        Iab,Iac,Icb be non empty compact Subset of REAL;
    assume that
A1: a <= c <= b and
A2: Iab = [.a,b.] and
A3: Iac = [.a,c.] and
A4: Icb = [.c,b.];
    let Dac be Division of Iac,
        Dcb be Division of Icb;
    assume
A5: c < Dcb.1;
    set Dacb = Dac ^ Dcb;
    for e1,e2 being ExtReal st e1 in dom Dacb & e2 in dom Dacb & e1 < e2 holds
    Dacb.e1 < Dacb.e2
    proof
      let e1,e2 be ExtReal;
      assume that
A6:   e1 in dom Dacb and
A7:   e2 in dom Dacb and
A8:   e1 < e2;
      per cases by A6,A7,FINSEQ_1:25;
        suppose
A9:       e1 in dom Dac & e2 in dom Dac; then
A10:      Dacb.e1 = Dac.e1 & Dacb.e2 = Dac.e2 by FINSEQ_1:def 7;
          e1 < e2 & Dac is increasing by A8;
          hence thesis by A9,A10;
        end;
        suppose
A11:      e1 in dom Dac & ex n be Nat st n in dom Dcb &
          e2 = len Dac + n;
          then consider n0 be Nat such that
A12:      n0 in dom Dcb and
A13:      e2 = len Dac + n0;
          Dacb.e1 = Dac.e1 & Dacb.e2 = Dcb.n0 by A11,A13,FINSEQ_1:def 7;
          hence thesis by A11,A12,A3,A4,A5,Th44,A1;
        end;
        suppose
A14:      (ex n be Nat st n in dom Dcb & e1 = len Dac + n) &
          e2 in dom Dac;
          then consider n0 be Nat such that
          n0 in dom Dcb and
A15:      e1 = len Dac + n0;
A16:      len Dac <= e1 by A15,NAT_1:11;
          e2 in Seg len Dac by FINSEQ_1:def 3,A14;
          then e2 <= len Dac by FINSEQ_1:1;
          hence thesis by A8,A16,XXREAL_0:2;
        end;
        suppose
A17:      (ex n be Nat st n in dom Dcb & e1 = len Dac + n) &
          (ex n be Nat st n in dom Dcb & e2 = len Dac + n);
          then consider n1 be Nat such that
A18:      n1 in dom Dcb and
A19:      e1 = len Dac + n1;
          consider n2 be Nat such that
A20:      n2 in dom Dcb and
A21:      e2 = len Dac + n2 by A17;
A22:      len Dac + n1 - len Dac < len Dac + n2 - len Dac
            by A8,A19,A21,XREAL_1:14;
          Dcb.n1 = Dacb.e1 & Dcb.n2 = Dacb.e2
          by A17,A19,A21,FINSEQ_1:def 7;
          hence thesis by A22,A18,A20,VALUED_0:def 13;
        end;
      end; then
A23:  Dacb is increasing;
A24:  rng Dacb c= Iab
      proof
        let x be object;
        assume
A25:    x in rng Dacb;
A26:    rng Dac c= [.a,c.] by A3,INTEGRA1:def 2;
        rng Dcb c= [.c,b.] by A4,INTEGRA1:def 2;
        then (rng Dac) \/ (rng Dcb) c= [.a,c.] \/ [.c,b.] by XBOOLE_1:13,A26;
        then rng Dacb c= [.a,c.] \/ [.c,b.] by FINSEQ_1:31;
        then rng Dacb c= [.a,b.] by A1,XXREAL_1:165;
        hence thesis by A25,A2;
      end;
      Dacb.(len Dacb) = upper_bound Iab
      proof
A27:    a <= b by A1,XXREAL_0:2;
A28:    len Dcb in Seg len Dcb by FINSEQ_1:3;
A29:    Seg len Dcb = dom Dcb by FINSEQ_1:def 3;
        len Dacb = len Dac + len Dcb by FINSEQ_1:22;
        then Dacb.(len Dacb) = Dcb.(len Dcb) by A29,A28,FINSEQ_1:def 7
                            .= upper_bound Icb by INTEGRA1:def 2
                            .= b by JORDAN5A:19,A1,A4
                            .= upper_bound [.a,b.] by A27,JORDAN5A:19;
        hence thesis by A2;
      end;
      hence thesis by A23,A24,INTEGRA1:def 2;
    end;
