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

theorem Th39:
  for f being non constant standard special_circular_sequence,
      i1,i2 st 1<=i1 & i1<i2 & i2+1<=len f holds
        mid(f,i1,i2) is being_S-Seq
proof
  let f be non constant standard special_circular_sequence, i1,i2;
  assume that
A1: 1<=i1 and
A2: i1<i2 and
A3: i2+1<=len f;
A4: mid(f,i1,i2)=(f/^(i1-'1))|(i2-'i1+1) by A2,FINSEQ_6:def 3;
  i1<i1+1 by NAT_1:13;
  then i1-1<i1+1-1 by XREAL_1:9;
  then i1-1<i2 by A2,XXREAL_0:2;
  then
A5: i1-'1<i2 by A1,XREAL_1:233;
  i2-'i1+1=i2-i1+1 by A2,XREAL_1:233
    .=i2-(i1-1)
    .=i2-(i1-'1) by A1,XREAL_1:233
    .=i2-'(i1-'1) by A5,XREAL_1:233;
  then
A6: mid(f,i1,i2)=(f|i2)/^(i1-'1) by A4,FINSEQ_5:80;
  i2<len f by A3,NAT_1:13;
  then i1-1+1<len (f|i2) by A2,FINSEQ_1:59;
  then
A7: i1-'1+1<len (f|i2) by A1,XREAL_1:233;
  1<i2 by A1,A2,XXREAL_0:2;
  hence thesis by A3,A6,A7,Th7,Th37;
end;
