reserve GX,GY for non empty TopSpace,
  x,y for Point of GX,
  r,s for Real;
reserve Q,P1,P2 for Subset of TOP-REAL 2;
reserve P for Subset of TOP-REAL 2;
reserve w1,w2 for Point of TOP-REAL 2;
reserve pa,pb for Point of TOP-REAL 2,
  s1,t1,s2,t2,s,t,s3,t3,s4,t4,s5,t5,s6,t6, l,sa,sd,ta,td for Real;
reserve s1a,t1a,s2a,t2a,s3a,t3a,sb,tb,sc,tc for Real;

theorem
  for p,q being Point of TOP-REAL 2 holds LSeg(p,q) \ {p,q} is convex
proof
  let p,q,w1,w2 be Point of TOP-REAL 2;
  set P = LSeg(p,q) \ {p,q};
  assume that
A1: w1 in P and
A2: w2 in P;
A3: w1 in LSeg(p,q) by A1,XBOOLE_0:def 5;
A4: w2 in LSeg(p,q) by A2,XBOOLE_0:def 5;
A5: not w1 in {p,q} by A1,XBOOLE_0:def 5;
A6: not w2 in {p,q} by A2,XBOOLE_0:def 5;
A7: w1 <> p by A5,TARSKI:def 2;
A8: w2 <> p by A6,TARSKI:def 2;
A9: w1 <> q by A5,TARSKI:def 2;
A10: w2 <> q by A6,TARSKI:def 2;
A11: not p in LSeg(w1,w2) by A3,A4,A7,A8,SPPOL_1:7,TOPREAL1:6;
  not q in LSeg(w1,w2) by A3,A4,A9,A10,SPPOL_1:7,TOPREAL1:6;
  then LSeg(w1,w2) misses {p,q} by A11,ZFMISC_1:51;
  hence thesis by A3,A4,TOPREAL1:6,XBOOLE_1:86;
end;
