reserve D for non empty set,
  D1,D2,x,y for set,
  n,k for Nat,
  p,x1 ,r for Real,
  f for Function;
reserve F for Functional_Sequence of D1,D2;
reserve G,H,H1,H2,J for Functional_Sequence of D,REAL;
reserve x for Element of D,
  X,Y for set,
  S1,S2 for Real_Sequence,
  f for PartFunc of D,REAL;

theorem Th42:
  H is_unif_conv_on X iff X common_on_dom H & H is_point_conv_on X
& for r st 0<r ex k st for n,x st n>=k & x in X holds |.(H.n).x-(lim(H,X)).x.|
  <r
proof
  thus H is_unif_conv_on X implies X common_on_dom H & H is_point_conv_on X &
for r st 0<r ex k st for n,x st n>=k & x in X holds |.(H.n).x-(lim(H,X)).x.|<r
  proof
    assume
A1: H is_unif_conv_on X;
    then consider f such that
A2: X = dom f and
A3: for p st p>0 ex k st for n,x st n>= k & x in X holds |.(H.n).x-f
    .x .|<p;
    thus X common_on_dom H by A1;
A4: now
      let x such that
A5:   x in X;
      let p;
      assume p>0;
      then consider k such that
A6:   for n,x st n>=k & x in X holds |.(H.n).x-f.x.|<p by A3;
      take k;
      let n;
      assume n>=k;
      hence |.(H.n).x-f.x.|<p by A5,A6;
    end;
    thus H is_point_conv_on X by A1,Th21;
    then
A7: f = (lim(H,X)) by A2,A4,Th20;
    let r;
    assume r > 0;
    then consider k such that
A8: for n,x st n>=k & x in X holds |.(H.n).x-f.x.|<r by A3;
    take k;
    let n,x;
    assume n>=k & x in X;
    hence thesis by A7,A8;
  end;
  assume that
A9: X common_on_dom H and
A10: H is_point_conv_on X and
A11: for r st 0<r ex k st for n,x st n>=k & x in X holds |.(H.n).x-(
  lim(H,X)).x.|<r;
  dom lim(H,X) = X by A10,Def13;
  hence thesis by A9,A11;
end;
