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

theorem Th40:
  for f being non constant standard special_circular_sequence,
      i1,i2 st 1<i1 & i1<i2 & i2<=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<=len f;
A4: i2-(i1-'1)<=len f-(i1-'1) by A3,XREAL_1:9;
  1+1<=i1 by A1,NAT_1:13;
  then 1+1-1<=i1-1 by XREAL_1:9;
  then
A5: 1<=i1-'1 by NAT_D:39;
A6: mid(f,i1,i2)=(f/^(i1-'1))|(i2-'i1+1) by A2,FINSEQ_6:def 3;
A7: i2-i1>0 by A2,XREAL_1:50;
  then
A8: 1<>i2-'i1+1 by XREAL_0:def 2;
  i2-'i1>0 by A7,XREAL_0:def 2;
  then
A9: i2-'i1>=0+1 by NAT_1:13;
  then
A10: 1<=i2-'i1+1 by NAT_1:13;
  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 i1-'1<i2 by A1,XREAL_1:233;
  then
A11: i1-'1<len f by A3,XXREAL_0:2;
  then len (f/^(i1-'1))=len f-(i1-'1) by RFINSEQ:def 1;
  then i2-(i1-1)<=len (f/^(i1-'1)) by A1,A4,XREAL_1:233;
  then i2-i1+1<=len (f/^(i1-'1));
  then
A12: i2-'i1+1<=len (f/^(i1-'1)) by A9,NAT_D:39;
A13: i1<len f by A2,A3,XXREAL_0:2;
  then i1-'1+1<len f by A1,XREAL_1:235;
  then i1-'1+1-(i1-'1)<len f-(i1-'1) by XREAL_1:9;
  then
A14: 1<=len (f/^(i1-'1)) by A11,RFINSEQ:def 1;
  i1-'1+1<len f by A1,A13,XREAL_1:235;
  then mid(f/^(i1-'1),1,i2-'i1+1) is being_S-Seq by A5,A14,A10,A12,A8,Th38,
JORDAN3:6;
  hence thesis by A6,A10,FINSEQ_6:116;
end;
