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 Th74:
  for S being functional with_common_domain set holds S c= product product" S
proof
  let S be functional with_common_domain set;
  let f be object;
  assume
A1: f in S;
  then reconsider f as Element of S;
A2: dom f = DOM S by A1,Lm2
    .= dom product" S by Def12;
  for i being object st i in dom product" S holds f.i in (product" S).i
  proof
    let i be object;
    assume i in dom product" S;
    then (product" S).i = pi(S,i) by Def12;
    hence thesis by A1,Def6;
  end;
  hence thesis by A2,Th9;
end;
