reserve k, l, m, n, i, j for Nat,
  K, N for non empty Subset of NAT,
  Ke, Ne, Me for Subset of NAT,
  X,Y for set;
reserve f for Function of Segm n,Segm k;
reserve x,y for set;

theorem Th57:
  for X,Y for x,y being object
   st (Y is empty implies X is empty) & not x in X
for
  F be Function of X,Y ex G be Function of X\/{x},Y\/{y} st G|X = F & G.x=y
proof
  let X,Y;
  let x,y be object such that
A1: Y is empty implies X is empty and
A2: not x in X;
  set Y1=Y\/{y};
  set X1=X\/{x};
  deffunc F(set)=y;
  let F be Function of X,Y;
  y in {y} by TARSKI:def 1;
  then
A3: for x9 be set st x9 in X1\X holds F(x9) in Y1 by XBOOLE_0:def 3;
A4: X c= X1 & Y c= Y1 by XBOOLE_1:7;
  consider G be Function of X1,Y1 such that
A5: G|X = F & for x9 be set st x9 in X1\X holds G.x9 = F(x9) from Sch2(
  A3,A4,A1);
  x in {x} by TARSKI:def 1;
  then x in X1 by XBOOLE_0:def 3;
  then x in X1\X by A2,XBOOLE_0:def 5;
  then G.x=y by A5;
  hence thesis by A5;
end;
