reserve r,s,t,u for Real;

theorem Th57:
  for X being LinearTopSpace, K being compact Subset of X, C being
closed Subset of X st K misses C ex V being a_neighborhood of 0.X st K+V misses
  C+V
proof
  let X be LinearTopSpace, K be compact Subset of X, C be closed Subset of X
  such that
A1: K misses C;
  per cases;
  suppose
A2: K = {};
    take V = [#]X;
    thus V is a_neighborhood of 0.X by TOPGRP_1:21;
    K+V = {} by A2,CONVEX1:40;
    hence thesis;
  end;
  suppose
A3: K <> {};
    set xV = { [x,Vx] where x is Point of X, Vx is open a_neighborhood of 0.X:
    x in K & Vx is symmetric & x+Vx+Vx+Vx misses C};
A4: now
      let x be Point of X such that
A5:   x in K;
      -x+C` = {-x + u where u is Point of X: u in C`} & K c= C` by A1,
RUSUB_4:def 8,SUBSET_1:23;
      then -x+x in -x+C` by A5;
      then 0.X in -x+C` by RLVECT_1:5;
      then -x+C` is a_neighborhood of 0.X by CONNSP_2:3;
      then consider V9 being open a_neighborhood of 0.X such that
      V9 is symmetric and
A6:   V9+V9 c= -x+C` by Th56;
      consider Vx being open a_neighborhood of 0.X such that
A7:   Vx is symmetric and
A8:   Vx+Vx c= V9 by Th56;
      take Vx;
      thus Vx is symmetric by A7;
      Vx c= V9
      proof
        let z be object;
        assume
A9:     z in Vx;
        then reconsider z as Point of X;
        0.X in Vx by CONNSP_2:4;
        then z+0.X in Vx+Vx by A9,Th3;
        then z in Vx+Vx;
        hence thesis by A8;
      end;
      then Vx+(Vx+Vx) c= V9 + V9 by A8,Th11;
      then Vx+Vx+Vx c=-x+C` by A6;
      then x+(Vx+Vx+Vx) c= x+(-x+C`) by Th8;
      then x+Vx+(Vx+Vx) c= x+(-x+C`) by Th7;
      then x+Vx+Vx+Vx c= x+(-x+C`) by CONVEX1:36;
      then x+Vx+Vx+Vx c= x+-x+C` by Th6;
      then x+Vx+Vx+Vx c= 0.X+C` by RLVECT_1:def 10;
      then x+Vx+Vx+Vx c= C` by Th5;
      hence x+Vx+Vx+Vx misses C by SUBSET_1:23;
    end;
A10: now
      consider x be object such that
A11:  x in K by A3,XBOOLE_0:def 1;
      reconsider x as Point of X by A11;
      consider Vx being open a_neighborhood of 0.X such that
A12:  Vx is symmetric & x+Vx+Vx+Vx misses C by A4,A11;
      take z = [x,Vx];
      thus z in xV by A11,A12;
    end;
    defpred P[object,object] means
ex v1,v2 being Point of X, V1,V2 being open
    a_neighborhood of 0.X st $1 = [v1,V1] & $2 = [v2,V2] & v1+V1 = v2+V2;
A13: for x,y,z being object st P[x,y] & P[y,z] holds P[x,z]
    proof
      let x,y,z be object;
      given v1,v2 being Point of X, V1,V2 being open a_neighborhood of 0.X
      such that
A14:  x = [v1,V1] and
A15:  y = [v2,V2] and
A16:  v1+V1 = v2+V2;
      given w1,w2 being Point of X, W1,W2 being open a_neighborhood of 0.X
      such that
A17:  y = [w1,W1] and
A18:  z = [w2,W2] & w1+W1 = w2+W2;
      take v1,w2,V1,W2;
      v2 = w1 by A15,A17,XTUPLE_0:1;
      hence thesis by A14,A15,A16,A17,A18,XTUPLE_0:1;
    end;
    reconsider xV as non empty set by A10;
A19: for x being object st x in xV holds P[x,x]
    proof
      let x be object;
      assume x in xV;
      then
      ex v being Point of X, V being open a_neighborhood of 0.X st x = [v
      ,V] & v in K & V is symmetric & v+V+V+V misses C;
      hence thesis;
    end;
A20: for x,y being object st P[x,y] holds P[y,x];
    consider eqR being Equivalence_Relation of xV such that
A21: for x,y being object holds [x,y] in eqR iff x in xV & y in xV & P[x,
    y] from EQREL_1:sch 1(A19,A20,A13);
    now
      let X be set;
      assume X in Class eqR;
      then ex x being object st x in xV & X = Class(eqR,x) by EQREL_1:def 3;
      hence X <> {} by EQREL_1:20;
    end;
    then consider g being Function such that
A22: dom g = Class eqR and
A23: for X being set st X in Class eqR holds g.X in X by FUNCT_1:111;
    set xVV = rng g;
    set F = { x+Vx where x is Point of X, Vx is open a_neighborhood of 0.X: [x
    ,Vx] in xVV};
    F c= bool the carrier of X
    proof
      let A be object;
      assume A in F;
      then
      ex x being Point of X, Vx being open a_neighborhood of 0.X st A = x
      +Vx & [x,Vx] in xVV;
      hence thesis;
    end;
    then reconsider F as Subset-Family of X;
A24: F is Cover of K
    proof
      let x be object;
      assume
A25:  x in K;
      then reconsider x as Point of X;
      consider Vx being open a_neighborhood of 0.X such that
A26:  Vx is symmetric & x+Vx+Vx+Vx misses C by A4,A25;
      [x,Vx] in xV by A25,A26;
      then
A27:  Class(eqR,[x,Vx]) in Class eqR by EQREL_1:def 3;
      then
A28:  g.Class(eqR,[x,Vx]) in xVV by A22,FUNCT_1:def 3;
      x+0.X in x+Vx by Lm1,CONNSP_2:4;
      then
A29:  x in x+Vx;
      g.Class(eqR,[x,Vx]) in Class(eqR,[x,Vx]) by A23,A27;
      then [g.Class(eqR,[x,Vx]),[x,Vx]] in eqR by EQREL_1:19;
      then consider
      v1,v2 being Point of X, V1,V2 being open a_neighborhood of 0.X
      such that
A30:  g.Class(eqR,[x,Vx]) = [v1,V1] and
A31:  [x,Vx] = [v2,V2] and
A32:  v1+V1 = v2+V2 by A21;
      x = v2 & Vx = V2 by A31,XTUPLE_0:1;
      then x+Vx in F by A30,A32,A28;
      hence thesis by A29,TARSKI:def 4;
    end;
    F is open
    proof
      let P be Subset of X;
      assume P in F;
      then
      ex x being Point of X, Vx being open a_neighborhood of 0.X st P = x
      +Vx & [x,Vx] in xVV;
      hence thesis;
    end;
    then consider G being Subset-Family of X such that
A33: G c= F and
A34: G is Cover of K and
A35: G is finite by A24,COMPTS_1:def 4;
    set FVx = {Vx where Vx is open a_neighborhood of 0.X: ex x being Point of
    X st x+Vx in G & [x,Vx] in xVV};
    FVx c= bool the carrier of X
    proof
      let A be object;
      assume A in FVx;
      then ex Vx being open a_neighborhood of 0.X st A = Vx & ex x being
      Point of X st x+Vx in G & [x,Vx] in xVV;
      hence thesis;
    end;
    then reconsider FVx as Subset-Family of X;
    defpred P[object,object] means ex x being Point of X, Vx being open
    a_neighborhood of 0.X st $1 = x+Vx & x+Vx in G & [x,Vx] in xVV & $2 = Vx;
A36: for x being object st x in G ex y being object st y in FVx & P[x,y]
    proof
      let x be object;
      assume
A37:  x in G;
      then x in F by A33;
      then consider
      z being Point of X, Vz being open a_neighborhood of 0.X such
      that
A38:  x = z+Vz & [z,Vz] in xVV;
      take Vz;
      thus thesis by A37,A38;
    end;
    consider f being Function of G,FVx such that
A39: for x being object st x in G holds P[x,f.x] from FUNCT_2:sch 1(A36
    );
    set FxVxVx = {x+Vx+Vx where x is Point of X, Vx is open a_neighborhood of
    0.X : x+Vx in G & [x,Vx] in xVV};
    take V = meet FVx;
A40: rng g c= xV
    proof
      let x be object;
      assume x in rng g;
      then consider y being object such that
A41:  y in dom g and
A42:  x = g.y by FUNCT_1:def 3;
      reconsider y as set by TARSKI:1;
      x in y by A22,A23,A41,A42;
      hence thesis by A22,A41;
    end;
A43: for A being Subset of X st A in G ex x being Point of X, Vx being
    open a_neighborhood of 0.X st A = x+Vx & [x,Vx] in xVV
    proof
      let A be Subset of X;
      assume A in G;
      then A in F by A33;
      hence thesis;
    end;
A44: now
      consider y be Point of X such that
A45:  y in K by A3,SUBSET_1:4;
      consider W being Subset of X such that
      y in W and
A46:  W in G by A34,A45,Th2;
      consider x being Point of X, Vx being open a_neighborhood of 0.X such
      that
A47:  W = x+Vx & [x,Vx] in xVV by A43,A46;
      Vx in FVx by A46,A47;
      hence ex z being set st z in FVx;
    end;
    then
A48: dom f = G by FUNCT_2:def 1;
A49: FVx c= rng f
    proof
      let z be object;
      assume z in FVx;
      then consider Vx being open a_neighborhood of 0.X such that
A50:  z = Vx and
A51:  ex x being Point of X st x+Vx in G & [x,Vx] in xVV;
      consider x being Point of X such that
A52:  x+Vx in G and
A53:  [x,Vx] in xVV by A51;
      consider v being Point of X, Vv being open a_neighborhood of 0.X such
      that
A54:  x+Vx = v+Vv and
      v+Vv in G and
A55:  [v,Vv] in xVV and
A56:  f.(x+Vx) = Vv by A39,A52;
      [[x,Vx],[v,Vv]] in eqR by A21,A40,A53,A54,A55;
      then [x,Vx] in Class(eqR,[v,Vv]) by EQREL_1:19;
      then
A57:  Class(eqR,[v,Vv]) = Class(eqR,[x,Vx]) by A40,A55,EQREL_1:23;
      consider A being object such that
A58:  A in dom g and
A59:  [x,Vx] = g.A by A53,FUNCT_1:def 3;
      consider a being object such that
A60:  a in xV and
A61:  A = Class(eqR,a) by A22,A58,EQREL_1:def 3;
      [x,Vx] in Class(eqR,a) by A22,A23,A58,A59,A61;
      then
A62:  Class(eqR,a) = Class(eqR,[x,Vx]) by A60,EQREL_1:23;
      consider B being object such that
A63:  B in dom g and
A64:  [v,Vv] = g.B by A55,FUNCT_1:def 3;
      consider b being object such that
A65:  b in xV and
A66:  B = Class(eqR,b) by A22,A63,EQREL_1:def 3;
      [v,Vv] in Class(eqR,b) by A22,A23,A63,A64,A66;
      then [x,Vx] = [v,Vv] by A57,A59,A64,A61,A65,A66,A62,EQREL_1:23;
      then Vx = Vv by XTUPLE_0:1;
      hence thesis by A48,A50,A52,A56,FUNCT_1:3;
    end;
A67: for x being Point of X, Vx being open a_neighborhood of 0.X st x+Vx
    in G & [x,Vx] in xVV holds x in K & Vx is symmetric & x+Vx+Vx+Vx misses C
    proof
      let x be Point of X, Vx be open a_neighborhood of 0.X such that
A68:  x+Vx in G and
A69:  [x,Vx] in xVV;
      consider A being object such that
A70:  A in dom g and
A71:  [x,Vx] = g.A by A69,FUNCT_1:def 3;
      consider a being object such that
A72:  a in xV and
A73:  A = Class(eqR,a) by A22,A70,EQREL_1:def 3;
      [x,Vx] in Class(eqR,a) by A22,A23,A70,A71,A73;
      then
A74:  Class(eqR,a) = Class(eqR,[x,Vx]) by A72,EQREL_1:23;
      x+Vx in F by A33,A68;
      then consider
      v being Point of X, Vv being open a_neighborhood of 0.X such
      that
A75:  x+Vx = v+Vv and
A76:  [v,Vv] in xVV;
      [[x,Vx],[v,Vv]] in eqR by A21,A40,A69,A75,A76;
      then [x,Vx] in Class(eqR,[v,Vv]) by EQREL_1:19;
      then
A77:  Class(eqR,[v,Vv]) = Class(eqR,[x,Vx]) by A40,A76,EQREL_1:23;
      consider B being object such that
A78:  B in dom g and
A79:  [v,Vv] = g.B by A76,FUNCT_1:def 3;
      consider b being object such that
A80:  b in xV and
A81:  B = Class(eqR,b) by A22,A78,EQREL_1:def 3;
      [v,Vv] in Class(eqR,b) by A22,A23,A78,A79,A81;
      then
A82:  [x,Vx] = [v,Vv] by A77,A71,A79,A73,A80,A81,A74,EQREL_1:23;
      then
A83:  Vx = Vv by XTUPLE_0:1;
      [v,Vv] in xV by A40,A76;
      then consider
      u being Point of X, Vu being open a_neighborhood of 0.X such
      that
A84:  [u,Vu] = [v,Vv] and
A85:  u in K & Vu is symmetric & u+Vu+Vu+Vu misses C;
      Vv = Vu by A84,XTUPLE_0:1;
      hence thesis by A84,A85,A82,A83,XTUPLE_0:1;
    end;
    now
      let Z be set;
      hereby
        reconsider A = C+V as set;
        assume Z in {{}};
        then
A86:    Z = {} by TARSKI:def 1;
        consider y be Point of X such that
A87:    y in K by A3,SUBSET_1:4;
        consider W being Subset of X such that
        y in W and
A88:    W in G by A34,A87,Th2;
        consider x being Point of X, Vx being open a_neighborhood of 0.X such
        that
A89:    W = x+Vx & [x,Vx] in xVV by A43,A88;
A90:    x+Vx+Vx+Vx misses C by A67,A88,A89;
        reconsider B = x+Vx+Vx as set;
        take A,B;
        thus A in {C+V} by TARSKI:def 1;
        thus B in FxVxVx by A88,A89;
A91:    Vx is symmetric by A67,A88,A89;
        now
A92:      C+V = {u + v where u,v is Point of X: u in C & v in V} by
RUSUB_4:def 9;
          assume A meets B;
          then consider z being object such that
A93:      z in A and
A94:      z in B by XBOOLE_0:3;
          reconsider z as Point of X by A93;
          consider u,v being Point of X such that
A95:      z = u+v and
A96:      u in C and
A97:      v in V by A93,A92;
          Vx in FVx by A88,A89;
          then v in Vx by A97,SETFAM_1:def 1;
          then -v in Vx by A91,Th25;
          then
A98:      z+-v in x+Vx+Vx+Vx by A94,Th3;
          z+-v = u+(v+-v) by A95,RLVECT_1:def 3
            .= u+0.X by RLVECT_1:5
            .= u;
          hence contradiction by A90,A96,A98,XBOOLE_0:3;
        end;
        hence Z = A /\ B by A86;
      end;
      given A,B being set such that
A99:  A in {C+V} and
A100: B in FxVxVx and
A101: Z = A /\ B;
A102: A = C+V by A99,TARSKI:def 1;
      consider x being Point of X, Vx being open a_neighborhood of 0.X such
      that
A103: B = x+Vx+Vx and
A104: x+Vx in G & [x,Vx] in xVV by A100;
A105: x+Vx+Vx+Vx misses C by A67,A104;
A106: Vx is symmetric by A67,A104;
      now
A107:   C+V = {u + v where u,v is Point of X: u in C & v in V} by RUSUB_4:def 9
;
        assume A meets B;
        then consider z being object such that
A108:   z in A and
A109:   z in B by XBOOLE_0:3;
        reconsider z as Point of X by A99,A108;
        consider u,v being Point of X such that
A110:   z = u+v and
A111:   u in C and
A112:   v in V by A102,A108,A107;
        Vx in FVx by A104;
        then v in Vx by A112,SETFAM_1:def 1;
        then -v in Vx by A106,Th25;
        then
A113:   z+-v in x+Vx+Vx+Vx by A103,A109,Th3;
        z+-v = u+(v+-v) by A110,RLVECT_1:def 3
          .= u+0.X by RLVECT_1:5
          .= u;
        hence contradiction by A105,A111,A113,XBOOLE_0:3;
      end;
      then A /\ B = {};
      hence Z in {{}} by A101,TARSKI:def 1;
    end;
    then INTERSECTION ({C+V}, FxVxVx) = {{}} by SETFAM_1:def 5;
    then union INTERSECTION ({C+V}, FxVxVx) = {} by ZFMISC_1:25;
    then (C+V) /\ union FxVxVx = {} by SETFAM_1:25;
    then
A114: C+V misses union FxVxVx;
A115: FVx is open
    proof
      let P be Subset of X;
      assume P in FVx;
      then ex Vx being open a_neighborhood of 0.X st P = Vx & ex x being
      Point of X st x+Vx in G& [x,Vx] in xVV;
      hence thesis;
    end;
    f"FVx is finite by A35,FINSET_1:1;
    then FVx is finite by A49,FINSET_1:9;
    then V is open by A115,TOPS_2:20;
    then
A116: Int V = V by TOPS_1:23;
    now
      let A be set;
      assume
A117: A in FVx;
      then reconsider A9=A as Subset of X;
      ex Vx being open a_neighborhood of 0.X st A = Vx & ex x being
      Point of X st x+Vx in G & [x,Vx] in xVV by A117;
      then Int A9 c= A9 & 0.X in Int A9 by CONNSP_2:def 1,TOPS_1:16;
      hence 0.X in A;
    end;
    then 0.X in V by A44,SETFAM_1:def 1;
    hence V is a_neighborhood of 0.X by A116,CONNSP_2:def 1;
    set FxVxV = {x+Vx+V where x is Point of X, Vx is open a_neighborhood of 0.
X:  x+Vx in G & [x,Vx] in xVV};
A118: union FxVxV c= union FxVxVx
    proof
      let z be object;
      assume z in union FxVxV;
      then consider Y being set such that
A119: z in Y and
A120: Y in FxVxV by TARSKI:def 4;
      consider x being Point of X, Vx being open a_neighborhood of 0.X such
      that
A121: Y = x+Vx+V and
A122: x+Vx in G & [x,Vx] in xVV by A120;
A123: x+Vx+Vx in FxVxVx by A122;
      x+Vx+V = {u + v where u,v is Point of X: u in x+Vx & v in V} by
RUSUB_4:def 9;
      then consider u,v being Point of X such that
A124: z = u+v and
A125: u in x+Vx and
A126: v in V by A119,A121;
      Vx in FVx by A122;
      then v in Vx by A126,SETFAM_1:def 1;
      then u+v in x+Vx+Vx by A125,Th3;
      hence thesis by A124,A123,TARSKI:def 4;
    end;
    K+V c= union FxVxV
    proof
      let z be object;
A127: K+V = {u + v where u,v is Point of X: u in K & v in V} by RUSUB_4:def 9;
      assume z in K+V;
      then consider u,v being Point of X such that
A128: z = u+v and
A129: u in K and
A130: v in V by A127;
      consider Vu being Subset of X such that
A131: u in Vu and
A132: Vu in G by A34,A129,Th2;
      consider x being Point of X, Vx being open a_neighborhood of 0.X such
      that
A133: Vu = x+Vx and
A134: [x,Vx] in xVV by A43,A132;
A135: x+Vx+V in FxVxV by A132,A133,A134;
      z in x+Vx+V by A128,A130,A131,A133,Th3;
      hence thesis by A135,TARSKI:def 4;
    end;
    hence thesis by A118,A114,XBOOLE_1:1,63;
  end;
end;
