reserve x,y,z,r,s for ExtReal;
reserve A,B for ext-real-membered set;
reserve A,B for ext-real-membered set;

theorem Th80:
  A is interval implies for x,y,r st x in A & y in A & x <= r & r
  <= y holds r in A
proof
  assume
A1: A is interval;
  let x,y,r such that
A2: x in A and
A3: y in A and
A4: x <= r and
A5: r <= y;
A6: r in [.x,y.] by A4,A5,XXREAL_1:1;
  [.x,y.] c= A by A1,A2,A3;
  hence thesis by A6;
end;
