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 Th11:
  for t1,P st P = {|[ s,t ]|:t1<t } holds P is convex
proof
  let t1,P;
  assume
A1: P = { |[ s,t ]|:t1<t };
  let w1,w2 such that
A2: w1 in P and
A3: w2 in P;
  let x be object such that
A4: x in LSeg(w1,w2);
  consider s3,t3 such that
A5: |[ s3,t3 ]|=w1 and
A6: t1<t3 by A1,A2;
A7: w1`2=t3 by A5,EUCLID:52;
  consider s4,t4 such that
A8: |[ s4,t4 ]|=w2 and
A9: t1<t4 by A1,A3;
A10: w2`2=t4 by A8,EUCLID:52;
  consider l such that
A11: x = (1-l)*w1 + l*w2 and
A12: 0 <= l and
A13: l <= 1 by A4;
  set w = (1-l)*w1 + l*w2;
A14: w= |[((1-l)*w1)`1+(l*w2)`1,((1-l)*w1)`2+ (l*w2)`2]| by EUCLID:55;
A15: (1-l)*w1=|[ (1-l)*w1`1 ,(1-l)*w1`2 ]| by EUCLID:57;
A16: l*w2=|[ l*w2`1 ,l*w2`2 ]| by EUCLID:57;
A17: ((1-l)*w1)`2= (1-l)*w1`2 by A15,EUCLID:52;
  (l*w2)`2=l*w2`2 by A16,EUCLID:52;
  then w`2=(1-l)* w1`2+ l* w2`2 by A14,A17,EUCLID:52;
  then
A18: t1< w`2 by A6,A7,A9,A10,A12,A13,XREAL_1:175;
  w = |[w`1, w`2]| by EUCLID:53;
  hence thesis by A1,A11,A18;
end;
