  reserve n,m,i for Nat,
          p,q for Point of TOP-REAL n,
          r,s for Real,
          R for real-valued FinSequence;

theorem Th20:
  for X,Y be set for F be Function-yielding Function
    for x,y be object st F is Funcs(X,Y)-valued or y in dom <:F:>
  holds F.x.y = <:F:>.y.x
proof
  let X,Y be set;
  let F be Function-yielding Function;
  set FF=dom F;
A1: dom doms F= dom F by FUNCT_6:59;
  let x,y be object such that
A2:F is Funcs(X,Y)-valued or y in dom <:F:>;
  per cases by A2;
    suppose y in dom <:F:> & x in dom F;
      hence F.x.y = <:F:>.y.x by FUNCT_6:34;
    end;
    suppose
A3:   y in dom <:F:> & not x in dom F;
      then dom (<:F:>.y) = FF by FUNCT_6:31;
      then
A4:     <:F:>.y.x = {} by A3,FUNCT_1:def 2;
      F.x={} by A3,FUNCT_1:def 2;
      hence thesis by A4;
    end;
    suppose
A5:   not y in dom <:F:> & x in dom F & F is Funcs(X,Y)-valued;
      then
A6:   <:F:>.y ={} by FUNCT_1:def 2;
A7:   rng F c= Funcs(X,Y) by RELAT_1:def 19,A5;
      F.x in rng F by A5,FUNCT_1:def 3;
      then consider Fx be Function such that
A8:     Fx=F.x
      and
A9:    dom Fx=X
      and
        rng Fx c= Y by A7,FUNCT_2:def 2;
      now
        let x be object;
        assume
A10:      x in dom F;
        then F.x in rng F by FUNCT_1:def 3;
        then ex Fx be Function st Fx=F.x & dom Fx=X & rng Fx c= Y
          by A7,FUNCT_2:def 2;
        hence (doms F).x = X by A10,FUNCT_6:22;
      end;
      then doms F = dom F --> X by A1,FUNCOP_1:11;
      then dom <:F:> = meet (dom F --> X) by FUNCT_6:29;
      then dom <:F:> = X by A5,FUNCT_6:27;
      then Fx.y={} by A9,A5,FUNCT_1:def 2;
      hence thesis by A6,A8;
    end;
    suppose
A11:  not y in dom <:F:> & not x in dom F;
      then
A12:    F.x ={} by FUNCT_1:def 2;
A13:  {} .x={};
      <:F:>.y ={} by A11,FUNCT_1:def 2;
      hence thesis by A12,A13;
    end;
end;
