reserve x1, x2, x3, x4, x5, x6, x7 for set;

theorem Th76:
  for A being connected Subset of R^1, a, b being Real st
  a in A & b in A holds [.a,b.] c= A
proof
  let A be connected Subset of R^1, 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 ex z being Real st z = x & a <= z & z <= b;
  hence thesis by A1,A2,Th75;
end;
