reserve S,T,W,Y for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem Th7:
  for X,Y be RealNormSpace,
        Z be Subset of [:X,Y:],
      x,y be object
  holds
    [x,y] in Z iff [y,x] in (Exch(Y,X)) " Z
  proof
    let X,Y be RealNormSpace,
          Z be Subset of [:X,Y:],
        x,y be object;

    set I = Exch(Y,X);

    hereby
      assume
      A1: [x,y] in Z; then
      consider x1 be Point of X, y1 be Point of Y such that
      A2: [x,y] = [x1,y1] by PRVECT_3:18;
      A3: x = x1 & y = y1 by A2,XTUPLE_0:1;
      A4: [x,y] = I.(y1,x1) by A2,Def1
               .= I.[y,x] by A3;

      [y1,x1] in the carrier of [:Y,X:];
      hence [y,x] in I"Z by A1,A3,A4,FUNCT_2:38;
    end;

    assume
    A5: [y,x] in I"Z; then
    consider y1 be Point of Y, x1 be Point of X such that
    A6: [y,x] = [y1,x1] by PRVECT_3:18;
    A7: x = x1 & y = y1 by A6,XTUPLE_0:1;
    I.[y,x] = I.(y1,x1) by A6
            .= [x,y] by A7,Def1;
    hence [x,y] in Z by A5,FUNCT_2:38;
  end;
