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 Th73:
  for S being non empty functional set,
  i being set st i in dom product" S holds
  (product" S).i = the set of all f.i where f is Element of S
proof
  let S be non empty functional set, i be set;
  assume
A1: i in dom product" S;
  hereby
    let x be object;
    assume x in (product" S).i;
    then x in pi(S,i) by A1,Def12;
    then ex f being Function st f in S & x = f.i by Def6;
    hence x in the set of all f.i where f is Element of S;
  end;
  let x be object;
  assume x in the set of all f.i where f is Element of S;
  then ex f being Element of S st x = f.i;
  then x in pi(S,i) by Def6;
  hence thesis by A1,Def12;
end;
