reserve i, i1, i2, j, k for Nat,
  r, s for Real;
reserve D for non empty set,
  f1 for FinSequence of D;

theorem Th44:
  for f being non constant standard special_circular_sequence,
      g being FinSequence of TOP-REAL 2,i1,i2 being Nat st
    g is_a_part>_of f,i1,i2 & i1<i2 holds
      L~g is_S-P_arc_joining f/.i1,f/.i2
proof
  let f be non constant standard special_circular_sequence, g be FinSequence
  of TOP-REAL 2,i1,i2 be Nat;
  assume that
A1: g is_a_part>_of f,i1,i2 and
A2: i1<i2;
A3: 1<=i1 by A1;
  i1+1<=len f by A1;
  then
A4: i1<len f by NAT_1:13;
  then
A5: f/.i1=f.i1 by A3,FINSEQ_4:15;
A6: i2+1<=len f by A1;
  then
A7: i2<len f by NAT_1:13;
A8: 1<=len g by A1;
  then
A9: g/.len g=g.len g by FINSEQ_4:15;
A10: 1<=i2 by A1;
A11: g=mid(f,i1,i2) by A1,A2,Th25;
  then g.1=f.i1 by A3,A10,A4,A7,FINSEQ_6:118;
  then
A12: f/.i1=g/.1 by A8,A5,FINSEQ_4:15;
  g.len g=f.i2 by A1;
  then
A13: f/.i2=g/.len g by A10,A7,A9,FINSEQ_4:15;
  g is being_S-Seq by A2,A11,A3,A6,Th39;
  hence thesis by A12,A13,TOPREAL4:def 1;
end;
