
theorem Th19:
  for A being non empty connected Subset of I[01], a, b being Real
 st a in A & b in A holds [.a,b.] c= A
proof
  let A be non empty connected Subset of I[01], a, b be Real;
  assume that
A1: a in A and
A2: b in A;
  let x be object;
  assume x in [.a,b.];
  then x in { y where y is Real: a <= y & y <= b } by RCOMP_1:def 1;
  then consider z being Real such that
A3: z = x and
A4: a <= z and
A5: z <= b;
A6: 0 <= z by A1,A4,BORSUK_1:43;
  b <= 1 by A2,BORSUK_1:43;
  then z <= 1 by A5,XXREAL_0:2;
  then reconsider z as Point of I[01] by A6,BORSUK_1:43;
  z in A by A1,A2,A4,A5,Th18;
  hence thesis by A3;
end;
