reserve X for set;
reserve n,i for Element of NAT;
reserve a,b,c,d,e,r,x0 for Real;
reserve A for non empty closed_interval Subset of REAL;
reserve f,g,h for PartFunc of REAL,REAL n;
reserve E for Element of REAL n;

theorem Th1:
  a <= c & c <= b
  implies c in ['a,b'] & ['a,c'] c= ['a,b'] & ['c,b'] c= ['a,b']
  proof
    assume that
A1: a <= c and
A2: c <= b;
A3: c in [.a,b.] by A1,A2;
    hence c in ['a,b'] by A1,A2,INTEGRA5:def 3,XXREAL_0:2;
A4: ['c,b'] = [.c,b.] by A2,INTEGRA5:def 3;
    a <= b by A1,A2,XXREAL_0:2; then
A5: a in [.a,b.] & b in [.a,b.];
    ['a,b'] = [.a,b.] & ['a,c'] = [.a,c.]
    by A1,A2,INTEGRA5:def 3,XXREAL_0:2;
    hence thesis by A4,A3,A5,XXREAL_2:def 12;
  end;
