reserve a for set;
reserve L for lower-bounded sup-Semilattice;
reserve x for Element of L;
reserve L for complete LATTICE;

theorem Th47:
  for L being RelStr, AR being Relation of L
  holds AR is satisfying_SI implies AR is satisfying_INT
proof
  let L be RelStr, AR be Relation of L;
  assume
A1: AR is satisfying_SI;
  let x, z be Element of L;
  [x,z] in AR implies ex y be Element of L st [x,y] in AR & [y,z] in AR
  proof
    assume
A2: [x,z] in AR;
    per cases;
    suppose x <> z;
      then ex y be Element of L st ( [x,y] in AR)&( [y,z] in AR)&( x
      <> y) by A1,A2;
      hence thesis;
    end;
    suppose x = z;
      hence thesis by A2;
    end;
  end;
  hence thesis;
end;
