reserve p,q,r for FinSequence;
reserve u,v,x,y,y1,y2,z for object, A,D,X,Y for set;
reserve i,j,k,l,m,n for Nat;
reserve J for Nat;
reserve n for Nat;
reserve x,y,y1,y2,z,a,b for object, X,Y,Z,V1,V2 for set,
  f,g,h,h9,f1,f2 for Function,
  i for Nat,
  P for Permutation of X,
  D,D1,D2,D3 for non empty set,
  d1 for Element of D1,
  d2 for Element of D2,
  d3 for Element of D3;

theorem Th125:
  product <*D1,D2*> = the set of all <*d1,d2*>
proof
  thus product <*D1,D2*> c= the set of all <*d1,d2*>
  proof
    let a be object;
    assume a in product <*D1,D2*>;
    then ex x,y st x in D1 & y in D2 & a = <*x,y*> by Th122;
    hence thesis;
  end;
  let a be object;
  assume a in the set of all <*d1,d2*>;
  then ex d1,d2 st a = <*d1,d2*>;
  hence thesis by Th122;
end;
