reserve x,X,Y for set;
reserve g,r,r1,r2,p,p1,p2 for Real;
reserve R for Subset of REAL;
reserve seq,seq1,seq2,seq3 for Real_Sequence;
reserve Ns for increasing sequence of NAT;
reserve n for Nat;
reserve W for non empty set;
reserve h,h1,h2 for PartFunc of W,REAL;
reserve e1,e2 for ExtReal;
reserve h,h1,h2 for PartFunc of REAL,REAL;

theorem Th36:
  r in X /\ Y /\ dom (h1+h2) implies r in X /\ dom h1 & r in Y /\ dom h2
proof
  assume
A1: r in X /\ Y /\ dom (h1+h2);
  then r in dom (h1+h2) by XBOOLE_0:def 4;
  then r in dom h1 /\ dom h2 by VALUED_1:def 1;
  then
A2: r in dom h1 & r in dom h2 by XBOOLE_0:def 4;
  r in X /\ Y by A1,XBOOLE_0:def 4;
  then r in X & r in Y by XBOOLE_0:def 4;
  hence thesis by A2,XBOOLE_0:def 4;
end;
