reserve f,g,h for Function,
  A for set;
reserve F for Function,
  B,x,y,y1,y2,z for set;
reserve x,z for object;
reserve X for non empty set,
  Y for set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;
reserve Y for non empty set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;
reserve Y for set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;
reserve Y for non empty set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;
reserve Y for set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;
reserve Y for non empty set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;
reserve Y for set,
  F for BinOp of X,
  f,g,h for Function
  of Y,X,
  x,x1,x2 for Element of X;
reserve y for Element of Y;
reserve Y for non empty set,
  F for BinOp of X,
  f for Function of Y,X,
  x for Element of X,
  y for Element of Y;
reserve a,b,c for set;

theorem
  for a,b being object, f being Function holds a.-->b c= f iff a in dom f &
  f.a = b
proof
  let a,b be object, f be Function;
A1: dom(a.-->b) = {a};
A2: a in dom(a.-->b) by TARSKI:def 1;
  hereby
    assume
A3: a.-->b c= f;
    then {a} c= dom f by A1,GRFUNC_1:2;
    hence a in dom f by ZFMISC_1:31;
    thus f.a = (a.-->b).a by A2,A3,GRFUNC_1:2
      .= b by Th72;
  end;
  assume that
A4: a in dom f and
A5: f.a = b;
A6: now
    let x be object;
    assume x in dom(a.-->b);
    then x = a by TARSKI:def 1;
    hence (a.-->b).x = f.x by A5,Th72;
  end;
  dom(a.-->b) c= dom f by A4,ZFMISC_1:31;
  hence thesis by A6,GRFUNC_1:2;
end;
