reserve A,B,C for Ordinal,
  K,L,M,N for Cardinal,
  x,y,y1,y2,z,u for object,X,Y,Z,Z1,Z2 for set,
  n for Nat,
  f,f1,g,h for Function,
  Q,R for Relation;
reserve ff for Cardinal-Function;
reserve F,G for Cardinal-Function;
reserve A,B for set;

theorem Th66:
  g in sproduct f implies g|A in sproduct f|A
proof
A1: dom(g|A) = dom g /\ A by RELAT_1:61;
A2: dom(f|A) = dom f /\ A by RELAT_1:61;
  assume
A3: g in sproduct f;
  then
A4: dom(g|A) c= dom(f|A) by A1,A2,Th49,XBOOLE_1:26;
  now
    let x be object;
    assume
A5: x in dom(g|A);
    then
A6: (g|A).x = g.x by FUNCT_1:47;
A7: (f|A).x = f.x by A4,A5,FUNCT_1:47;
    x in dom g by A1,A5,XBOOLE_0:def 4;
    hence (g|A).x in (f|A).x by A3,A6,A7,Th49;
  end;
  hence thesis by A4,Def9;
end;
