reserve P for Subset of TOP-REAL 2,
  f,f1,f2,g for FinSequence of TOP-REAL 2,
  p,p1,p2,q,q1,q2 for Point of TOP-REAL 2,
  r1,r2,r19,r29 for Real,
  i,j,k,n for Nat;

theorem Th2:
  for i, j being Nat holds i+j = len f implies LSeg(f,i) = LSeg(Rev f,j)
proof
  let i, j be Nat;
  assume
A1: i+j = len f;
  per cases;
  suppose that
A2: 1 <= i and
A3: i + 1 <= len f;
A4: i+(j+1) = len f + 1 by A1;
A5: i in dom f by A2,A3,SEQ_4:134;
    then j+1 in dom Rev f by A4,FINSEQ_5:59;
    then
A6: j+1 <= len Rev f by FINSEQ_3:25;
A7: i+1+j = len f + 1 by A1;
A8: i+1 in dom f by A2,A3,SEQ_4:134;
    then j in dom Rev f by A7,FINSEQ_5:59;
    then
A9: 1 <= j by FINSEQ_3:25;
    thus LSeg(f,i) = LSeg(f/.i,f/.(i+1)) by A2,A3,TOPREAL1:def 3
      .= LSeg((Rev f)/.(j+1),f/.(i+1)) by A5,A4,FINSEQ_5:66
      .= LSeg((Rev f)/.j,(Rev f)/.(j+1)) by A8,A7,FINSEQ_5:66
      .= LSeg(Rev f,j) by A6,A9,TOPREAL1:def 3;
  end;
  suppose
A10: not 1 <= i;
    then i = 0 by NAT_1:14;
    then not j+1 <= len f by A1,NAT_1:13;
    then
A11: not j+1 <= len Rev f by FINSEQ_5:def 3;
    LSeg(f,i) = {} by A10,TOPREAL1:def 3;
    hence thesis by A11,TOPREAL1:def 3;
  end;
  suppose
A12: not i+1 <= len f;
    then
A13: LSeg(f,i) = {} by TOPREAL1:def 3;
    j < 1 by A1,A12,XREAL_1:6;
    hence thesis by A13,TOPREAL1:def 3;
  end;
end;
