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 Th2:
  a <= c & c <= d & d <= b & ['a,b'] c= X implies ['c,d'] c= X
  proof
    assume that
A1: a <= c & c <= d & d <= b and
A2: ['a,b'] c= X;
A3: ['c,d'] c= ['c,b'] by Th1,A1;
    c <= b by A1,XXREAL_0:2; then
    ['c,b'] c= ['a,b'] by Th1,A1; then
    ['c,d'] c= ['a,b'] by A3;
    hence thesis by A2;
  end;
