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

theorem Th43:
  for a,b,c being Real,Iac,Icb being non empty compact Subset of REAL st
  a <= c <= b & Iac=[.a,c.] & Icb=[.c,b.] holds
  for Dac being Division of Iac, Dcb being Division of Icb,
  i,j being Nat st i in dom Dac & j in dom Dcb holds
  (i < len Dac implies Dac.i < Dcb.j) & ((i = len Dac & c < Dcb.1) implies
  Dac.i < Dcb.j) & (Dcb.1 = c implies Dac.len Dac = Dcb.1)
  proof
    let a,b,c be Real,Iac,Icb be non empty compact Subset of REAL;
    assume that
A1: a <= c <= b and
A2: Iac=[.a,c.] and
A3: Icb=[.c,b.];
    let Dac be Division of Iac, Dcb be Division of Icb, i,j be Nat;
    assume that
A4: i in dom Dac and
A5: j in dom Dcb;
A6: Dac.(len Dac) = upper_bound Iac by INTEGRA1:def 2
                 .= c by A1,A2,JORDAN5A:19;
    rng Dcb c= [.c,b.] by A3,INTEGRA1:def 2;
    then Dcb.j in [.c,b.] by A5,FUNCT_1:3; then
A7: c <= Dcb.j by XXREAL_1:1;
    thus i < len Dac implies Dac.i < Dcb.j
    proof
      assume
A8:   i < len Dac;
      Seg len Dac = dom Dac by FINSEQ_1:def 3; then
      len Dac in dom Dac by FINSEQ_1:3;
      then Dac.i < c by A6,A8,A4,VALUED_0:def 13;
      hence Dac.i < Dcb.j by A7,XXREAL_0:2;
    end;
    thus i = len Dac & c < Dcb.1 implies Dac.i < Dcb.j
    proof
      assume
A9:   i = len Dac & c < Dcb.1;
A10:  1 in dom Dcb by FINSEQ_5:6;
      j in Seg len Dcb by A5,FINSEQ_1:def 3;
      then j = 1 or ... or j = len Dcb by FINSEQ_1:91;
      then per cases by XXREAL_0:1;
      suppose j = 1;
        hence thesis by A9,A6;
      end;
      suppose 1 < j;
        then Dcb.1 < Dcb.j by A10,A5, VALUED_0:def 13;
        hence thesis by A9,A6,XXREAL_0:2;
      end;
    end;
    thus Dcb.1 = c implies Dac.len Dac = Dcb.1 by A6;
  end;
