reserve a,b,p,x,x9,x1,x19,x2,y,y9,y1,y19,y2,z,z9,z1,z2 for object,
   X,X9,Y,Y9,Z,Z9 for set;
reserve A,D,D9 for non empty set;
reserve f,g,h for Function;
reserve A,B for set;
reserve x,y,i,j,k for object;

theorem
  z in dom f & f.z = x implies (f+~(x,y)).z = y
proof
  assume that
A1: z in dom f and
A2: f.z = x;
  f.z in dom(x.-->y) by A2,FUNCOP_1:74;
  then
A3: z in dom((x.-->y)*f) by A1,FUNCT_1:11;
  hence (f+~(x,y)).z = ((x.-->y)*f).z by Th13
    .= (x.-->y).x by A2,A3,FUNCT_1:12
    .= y by FUNCOP_1:72;
end;
