reserve a, b, c, d, r, s for Real,
  n for Element of NAT,
  p, p1, p2 for Point of TOP-REAL 2,
  x, y for Point of TOP-REAL n,
  C for Simple_closed_curve,
  A, B, P for Subset of TOP-REAL 2,
  U, V for Subset of (TOP-REAL 2)|C`,
  D for compact with_the_max_arc Subset of TOP-REAL 2;

theorem Th58:
  for o being Point of TOP-REAL 2 holds
  diffX2_1(o) is continuous Function of [:TOP-REAL 2, TOP-REAL 2:], R^1
proof
  let o be Point of TOP-REAL 2;
  reconsider Xo = diffX2_1(o) as Function of [:T2,T2:],R^1 by TOPMETR:17;
  for p being Point of [:T2,T2:], V being Subset of R^1
  st Xo.p in V & V is open holds
  ex W being Subset of [:T2,T2:] st p in W & W is open & Xo.:W c= V
  proof
    let p be Point of [:T2,T2:], V be Subset of R^1 such that
A1: Xo.p in V and
A2: V is open;
A3: Xo.p = p`2`1 - o`1 by Def1;
    set r = p`2`1 - o`1;
    reconsider V1 = V as open Subset of REAL by A2,BORSUK_5:39,TOPMETR:17;
    consider g being Real such that
A4: 0 < g and
A5: ].r-g,r+g.[ c= V1 by A1,A3,RCOMP_1:19;
    reconsider g as Element of REAL by XREAL_0:def 1;
    set W2 = {|[x,y]| where x, y is Real:
    p`2`1-g < x & x < p`2`1+g};
    W2 c= the carrier of T2
    proof
      let a be object;
      assume a in W2;
      then ex x, y being Real st
      a = |[x,y]| & p`2`1-g < x & x < p`2`1+g;
      hence thesis;
    end;
    then reconsider W2 as Subset of T2;
    take [:[#]T2,W2:];
A6: p`2 = |[p`2`1,p`2`2]| by EUCLID:53;
A7: p = [p`1,p`2] by Lm5,MCART_1:21;
A8: p`2`1-g < p`2`1-0 by A4,XREAL_1:15;
    p`2`1+0 < p`2`1+g by A4,XREAL_1:6;
    then p`2 in W2 by A6,A8;
    hence p in [:[#]T2,W2:] by A7,ZFMISC_1:def 2;
    W2 is open by PSCOMP_1:19;
    hence [:[#]T2,W2:] is open by BORSUK_1:6;
    let b be object;
    assume b in Xo.:[:[#]T2,W2:];
    then consider a being Point of [:T2,T2:] such that
A9: a in [:[#]T2,W2:] and
A10: Xo.a = b by FUNCT_2:65;
A11: a = [a`1,a`2] by Lm5,MCART_1:21;
A12: (diffX2_1(o)).a = a`2`1 - o`1 by Def1;
    a`2 in W2 by A9,A11,ZFMISC_1:87;
    then consider x2, y2 being Real such that
A13: a`2 = |[x2,y2]| and
A14: p`2`1-g < x2 and
A15: x2 < p`2`1+g;
A16: a`2`1 = x2 by A13,EUCLID:52;
    then
A17: p`2`1 - g - o`1 < a`2`1 - o`1 by A14,XREAL_1:9;
    a`2`1 - o`1 < p`2`1 + g - o`1 by A15,A16,XREAL_1:9;
    then a`2`1 - o`1 in ].r-g,r+g.[ by A17,XXREAL_1:4;
    hence thesis by A5,A10,A12;
  end;
  hence thesis by JGRAPH_2:10;
end;
