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
  x in dom f1 & y in dom f2 implies for y1,y2 holds
  [:f1,f2:].(x,y) = [y1,y2] iff (Frege<*f1,f2*>).<*x,y*> = <*y1,y2*>
proof
  assume
A1: x in dom f1 & y in dom f2;
  let y1,y2;
A2: <*f1.x,f2.y*>.1 = f1.x & <*f1.x,f2.y*>.2 = f2.y;
A3: <*y1,y2*>.1 = y1 & <*y1,y2*>.2 = y2;
  [f1.x,f2.y] = [y1,y2] iff f1.x = y1 & f2.y = y2 by XTUPLE_0:1;
  hence thesis by A1,A2,A3,Th142,FUNCT_3:def 8;
end;
