reserve C for Simple_closed_curve,
  p,q,p1 for Point of TOP-REAL 2,
  i,j,k,n for Nat,
  r,s for Real;

theorem
  for A,B being compact non empty Subset of TOP-REAL n,
  f being continuous RealMap of [:TOP-REAL n, TOP-REAL n:],
  g being RealMap of TOP-REAL n st for p being Point of TOP-REAL n
  ex G being Subset of REAL
  st G = { f.(p,q) where q is Point of TOP-REAL n : q in B } &
   g.p = lower_bound G
  holds lower_bound(f.:[:A,B:]) = lower_bound(g.:A)
proof
  let A,B be compact non empty Subset of TOP-REAL n;
  let f be continuous RealMap of [:TOP-REAL n, TOP-REAL n:];
  let g being RealMap of TOP-REAL n such that
A1: for p being Point of TOP-REAL n ex G being Subset of REAL st
  G = { f.(p,q) where q is Point of TOP-REAL n : q in B } &
   g.p = lower_bound G;
A2: [:A,B:] is compact by BORSUK_3:23;
  then
A3: f.:[:A,B:] is compact by Th1;
A4: f.:[:A,B:] is real-bounded by A2,Th1,RCOMP_1:10;
A5: g.:A c= f.:[:A,B:]
  proof
    let u be object;
    assume u in g.:A;
    then consider p being Point of TOP-REAL n such that
A6: p in A and
A7: u = g.p by FUNCT_2:65;
    consider q being Point of TOP-REAL n such that
A8: q in B by SUBSET_1:4;
    consider G being Subset of REAL such that
A9: G = { f.(p,q1) where q1 is Point of TOP-REAL n : q1 in B } and
A10: u = lower_bound G by A1,A7;
A11: f.(p,q) in G by A8,A9;
    G c= f.:[:A,B:]
    proof
      let u be object;
      assume u in G;
      then consider q1 being Point of TOP-REAL n such that
A12:  u = f.(p,q1) and
A13:  q1 in B by A9;
      [p,q1] in [:A,B:] by A6,A13,ZFMISC_1:87;
      hence thesis by A12,FUNCT_2:35;
    end;
    hence thesis by A3,A10,A11,Th2;
  end;
  then
A14: g.:A is bounded_below by A4,XXREAL_2:44;
A15: for r st r in f.:[:A,B:] holds lower_bound(g.:A)<=r
  proof
    let r;
    assume r in f.:[:A,B:];
    then consider pq being Point of [:TOP-REAL n, TOP-REAL n:] such that
A16: pq in [:A,B:] and
A17: r = f.pq by FUNCT_2:65;
    pq in the carrier of [:TOP-REAL n, TOP-REAL n:];
    then pq in [:the carrier of TOP-REAL n, the carrier of TOP-REAL n:]
    by BORSUK_1:def 2;
    then consider p,q being object such that
A18: p in the carrier of TOP-REAL n and
A19: q in the carrier of TOP-REAL n and
A20: pq = [p,q] by ZFMISC_1:84;
A21: q in B by A16,A20,ZFMISC_1:87;
    reconsider p,q as Point of TOP-REAL n by A18,A19;
    consider G being Subset of REAL such that
A22: G = { f.(p,q1) where q1 is Point of TOP-REAL n : q1 in B } and
A23: g.p = lower_bound G by A1;
A24: p in A by A16,A20,ZFMISC_1:87;
    G c= f.:[:A,B:]
    proof
      let u be object;
      assume u in G;
      then consider q1 being Point of TOP-REAL n such that
A25:  u = f.(p,q1) and
A26:  q1 in B by A22;
      [p,q1] in [:A,B:] by A24,A26,ZFMISC_1:87;
      hence thesis by A25,FUNCT_2:35;
    end;
    then
A27: G is bounded_below by A4,XXREAL_2:44;
    r = f.(p,q) by A17,A20;
    then r in G by A21,A22;
    then
A28: g.p <= r by A23,A27,SEQ_4:def 2;
    g.p in g.:A by A24,FUNCT_2:35;
    then lower_bound(g.:A)<=g.p by A14,SEQ_4:def 2;
    hence thesis by A28,XXREAL_0:2;
  end;
  for s st 0<s ex r st r in f.:[:A,B:] & r<lower_bound(g.:A)+s
  proof
    let s;
    assume 0<s;
    then consider r such that
A29: r in g.:A and
A30: r<lower_bound(g.:A)+s by A14,SEQ_4:def 2;
    take r;
    thus r in f.:[:A,B:] by A5,A29;
    thus thesis by A30;
  end;
  hence thesis by A4,A15,SEQ_4:def 2;
end;
