reserve p,p1,p2,p3,p11,p22,q,q1,q2 for Point of TOP-REAL 2,
  f,h for FinSequence of TOP-REAL 2,
  r,r1,r2,s,s1,s2 for Real,
  u,u1,u2,u5 for Point of Euclid 2,
  n,m,i,j,k for Nat,
  N for Nat,
  x,y,z for set;
reserve lambda for Real;

theorem Th17:
  j in dom(f|i) & j+1 in dom(f|i) implies LSeg(f,j) = LSeg(f|i,j)
proof
  assume that
A1: j in dom(f|i) and
A2: j+1 in dom(f|i);
A3: 1 <= j & j+1 <= len(f|i) by A1,A2,FINSEQ_3:25;
  set p1 = (f|i)/.j, p2 = (f|i)/.(j+1);
A4: f|i = f| (Seg i) by FINSEQ_1:def 16;
  then j in dom f /\ (Seg i) by A1,RELAT_1:61;
  then j in dom f by XBOOLE_0:def 4;
  then
A5: 1 <= j by FINSEQ_3:25;
  j+1 in dom f /\ (Seg i) by A2,A4,RELAT_1:61;
  then j+1 in dom f by XBOOLE_0:def 4;
  then
A6: j+1 <= len f by FINSEQ_3:25;
  p1 = f/.j & p2 = f/.(j+1) by A1,A2,FINSEQ_4:70;
  then LSeg(f,j) = LSeg(p1,p2) by A5,A6,TOPREAL1:def 3;
  hence thesis by A3,TOPREAL1:def 3;
end;
