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;
reserve A,B for Ordinal;
reserve n,k for Nat;

theorem
  for f,g being non-empty Function st f tolerates g
  for x being Element of product (f+*g) holds x|dom f in product f
proof
  let f,g be non-empty Function such that
A1: f tolerates g;
  let x be Element of product (f+*g);
A2: dom x = dom (f+*g) by Th9;
A3: dom (f+*g) = dom f \/ dom g by FUNCT_4:def 1;
  then
A4: dom f c= dom x by A2,XBOOLE_1:7;
A5: dom (x|dom f) = dom f by A2,A3,RELAT_1:62,XBOOLE_1:7;

now
    let z be object;
    assume
A6: z in dom (x|dom f);
    then
A7: (x|dom f).z = x.z by FUNCT_1:47;
    (f+*g).z = f.z by A1,A5,A6,FUNCT_4:15;
   hence (x|dom f).z in f.z by A2,A4,A5,A6,A7,Th9;
  end;
  hence thesis by A5,Th9;
end;
