 reserve h,h1 for 0-convergent non-zero Real_Sequence,
         c,c1 for constant Real_Sequence;

theorem Th3:
  for a,b be Real, I be Interval st a in I & b in I holds [.a,b.] c= I
proof
    let a,b be Real, I be Interval;
    assume that
A1:  a in I and
A2:  b in I;
    per cases by MEASURE5:1;
    suppose I is open_interval; then
     consider p,q be R_eal such that
A3:   I = ].p,q.[ by MEASURE5:def 2;
A4:  p < a & b < q by A1,A2,A3,XXREAL_1:4;
     now let x be Real;
      assume x in [.a,b.]; then
      a <= x & x <= b by XXREAL_1:1; then
      p < x & x < q by A4,XXREAL_0:2;
      hence x in I by A3,XXREAL_1:4;
     end;
     hence [.a,b.] c= I;
    end;
    suppose I is closed_interval; then
     consider p,q be Real such that
A5:   I = [.p,q.] by MEASURE5:def 3;
A6:  p <= a & b <= q by A1,A2,A5,XXREAL_1:1;
     now let x be Real;
      assume x in [.a,b.]; then
      a <= x & x <= b by XXREAL_1:1; then
      p <= x & x <= q by A6,XXREAL_0:2;
      hence x in I by A5,XXREAL_1:1;
     end;
     hence [.a,b.] c= I;
    end;
    suppose I is left_open_interval; then
     consider p be R_eal, q be Real such that
A7:   I = ].p,q.] by MEASURE5:def 5;
A8:  p < a & b <= q by A1,A2,A7,XXREAL_1:2;
     now let x be Real;
      assume x in [.a,b.]; then
      a <= x & x <= b by XXREAL_1:1; then
      p < x & x <= q by A8,XXREAL_0:2;
      hence x in I by A7,XXREAL_1:2;
     end;
     hence [.a,b.] c= I;
    end;
    suppose I is right_open_interval; then
     consider p be Real, q be R_eal such that
A9:   I = [.p,q.[ by MEASURE5:def 4;
A10:  p <= a & b < q by A1,A2,A9,XXREAL_1:3;
     now let x be Real;
      assume x in [.a,b.]; then
      a <= x & x <= b by XXREAL_1:1; then
      p <= x & x < q by A10,XXREAL_0:2;
      hence x in I by A9,XXREAL_1:3;
     end;
     hence [.a,b.] c= I;
    end;
end;
