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

theorem Th44:
  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
  c < Dcb.1 implies Dac.i < Dcb.j
  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 being Division of Iac, Dcb being Division of Icb, i,j being Nat;
    assume that
A4: i in dom Dac and
A5: j in dom Dcb;
    assume
A6: c < Dcb.1;
    i <= len Dac by A4,FINSEQ_3:25;
    then i < len Dac or i = len Dac by XXREAL_0:1;
    hence thesis by A6,A1,A2,A3,A4,A5,Th43;
  end;
