reserve i,k,n,m for Element of NAT;
reserve a,b,r,r1,r2,s,x,x1,x2 for Real;
reserve A for non empty closed_interval Subset of REAL;
reserve X for set;

theorem Th6:
  for f being PartFunc of REAL,REAL st A c= dom f holds f||A is
  total
proof
  let f be PartFunc of REAL,REAL;
  assume
A1: A c= dom f;
  dom (f||A) = dom f /\ A by RELAT_1:61
    .= A by A1,XBOOLE_1:28;
  hence thesis by PARTFUN1:def 2;
end;
