reserve x,y,z, X,Y,Z for set,
  n for Element of NAT;
reserve A for set,
  D for non empty set,
  a,b,c,l,r for Element of D,
  o,o9 for BinOp of D,
  f,g,h for Function of A,D;
reserve G for non empty multMagma;
reserve A for non empty set,
  a for Element of A,
  p for FinSequence of A,
  m1,m2 for Multiset of A;
reserve p,q for FinSequence of A;
reserve fm for Element of finite-MultiSet_over A;
reserve a,b,c for Element of D;

theorem Th45:
  for D1,D2,D being non empty set, f being Function of [:D1,D2:],D for
X1 being Subset of D1, X2 being Subset of D2, x1,x2 being set st x1 in X1 & x2
  in X2 holds f.(x1,x2) in (f.:^2).(X1,X2)
proof
  let D1,D2,D be non empty set, f be Function of [:D1,D2:],D;
  let X1 be Subset of D1, X2 be Subset of D2, x1,x2 be set;
  assume that
A1: x1 in X1 and
A2: x2 in X2;
  reconsider b = x2 as Element of D2 by A2;
  reconsider a = x1 as Element of D1 by A1;
A3: (f.:^2).(X1,X2) = f.:[:X1,X2:] & dom f = [:D1,D2:] by Th44,FUNCT_2:def 1;
  [a,b] in [:X1,X2:] by A1,A2,ZFMISC_1:87;
  hence thesis by A3,FUNCT_1:def 6;
end;
