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

theorem Th19:
  for f being FinSequence of TOP-REAL 2,i1,i2,i st 1<=i1 & i1<i2 &
  i2<=len f & 1<=i & i<i2-'i1+1 holds LSeg(mid(f,i1,i2),i)=LSeg(f,i+i1-'1)
proof
  let f be FinSequence of TOP-REAL 2,i1,i2,i;
  assume that
A1: 1<=i1 and
A2: i1<i2 and
A3: i2<=len f and
A4: 1<=i and
A5: i<i2-'i1+1;
A6: i+i1-'1+1=i+i1-1+1 by A1,NAT_D:37
    .=i+1+i1-1
    .=i+1+i1-'1 by A1,NAT_D:37;
  i2-i1+1=i2-'i1+1 by A2,XREAL_1:233;
  then i+i1<i2-i1+1+i1 by A5,XREAL_1:6;
  then i+i1+1-1<=i2+1-1 by NAT_1:13;
  then
A7: i+i1-1+1<=len f by A3,XXREAL_0:2;
  then
A8: i+i1-'1+1<=len f by A1,NAT_D:37;
  then
A9: i+i1-'1<len f by NAT_1:13;
  1+1<=i+i1 by A1,A4,XREAL_1:7;
  then 1+1-1<=i+i1-1 by XREAL_1:9;
  then
A10: 1<=i+i1-'1 by A1,NAT_D:37;
  then
A11: 1<i+i1-'1+1 by NAT_1:13;
A12: LSeg(f,i+i1-'1)=LSeg(f/.(i+i1-'1),f/.(i+i1-'1+1)) by A10,A8,TOPREAL1:def 3
;
A13: i+1<=i2-'i1+1 by A5,NAT_1:13;
A14: i+i1-'1+1<=len f by A1,A7,NAT_D:37;
A15: 1<=1+i by NAT_1:11;
A16: i<len mid(f,i1,i2) by A1,A2,A3,A5,FINSEQ_6:186;
  then
A17: mid(f,i1,i2)/.i=mid(f,i1,i2).i by A4,FINSEQ_4:15
    .=f.(i+i1-'1) by A1,A2,A3,A4,A5,FINSEQ_6:122
    .=f/.(i+i1-'1) by A10,A9,FINSEQ_4:15;
A18: i+1<=len mid(f,i1,i2) by A16,NAT_1:13;
  i+1<=len mid(f,i1,i2) by A16,NAT_1:13;
  then mid(f,i1,i2)/.(i+1)=mid(f,i1,i2).(i+1) by FINSEQ_4:15,NAT_1:11
    .=f.(i+1+i1-'1) by A1,A2,A3,A13,A15,FINSEQ_6:122
    .=f/.(i+1+i1-'1) by A11,A14,A6,FINSEQ_4:15;
  hence thesis by A4,A18,A12,A6,A17,TOPREAL1:def 3;
end;
