reserve i,j,k for Nat,
  r,s,r1,r2,s1,s2,sb,tb for Real,
  x for set,
  GX for non empty TopSpace;
reserve GZ for non empty TopSpace;
reserve f for non constant standard special_circular_sequence,
  G for non empty-yielding Matrix of TOP-REAL 2;
reserve G for non empty-yielding X_equal-in-line Y_equal-in-column Matrix of
  TOP-REAL 2;

theorem
  for G being Go-board holds i<=len G & j<=width G implies cell(G,i,j)=
  Cl Int cell(G,i,j)
proof
  let G be Go-board;
  set Y = Int cell(G,i,j);
  assume
A1: i<=len G & j<=width G;
A2: cell(G,i,j) c= Cl Y
  proof
    let x be object;
    assume
A3: x in cell(G,i,j);
    then reconsider p=x as Point of TOP-REAL 2;
    for G0 being Subset of TOP-REAL 2 st G0 is open holds p in G0 implies
    Y meets G0
    proof
      let G0 be Subset of TOP-REAL 2;
      assume
A4:   G0 is open;
      now
        reconsider u=p as Point of Euclid 2 by EUCLID:22;
        assume
A5:     p in G0;
A6:     j=0 or 0+1<=j by NAT_1:13;
        reconsider v=u as Element of REAL 2;
A7:     TopSpaceMetr Euclid 2 = the TopStruct of TOP-REAL 2 by EUCLID:def 8;
        then reconsider G00=G0 as Subset of TopSpaceMetr Euclid 2;
        G00 is open by A4,A7,PRE_TOPC:30;
        then consider r be Real such that
A8:     r>0 and
A9:     Ball(u,r) c= G00 by A5,TOPMETR:15;
        reconsider r as Real;
A10:    i=0 or 0+1<=i by NAT_1:13;
        now
          per cases by A1,A10,A6,XXREAL_0:1;
          case
A11:        i=0 & j=0;
            then
            p in { |[r2,s2]| : r2 <= G*(1,1)`1 & s2 <= G*(1,1)`2 } by A3,Th24;
            then consider r2,s2 such that
A12:        p=|[r2,s2]| and
A13:        r2 <= G*(1,1)`1 and
A14:        s2 <= G*(1,1)`2;
            set r3=r2-r/2 ,s3=s2-r/2;
A15:        r*2">0 by A8,XREAL_1:129;
            then s3<s3+r/2 by XREAL_1:29;
            then
A16:        s3<G*(1,1)`2 by A14,XXREAL_0:2;
            reconsider q0=|[r3,s3]| as Point of TOP-REAL 2;
            reconsider u0=q0 as Point of Euclid 2 by EUCLID:22;
            r3<r3+r/2 by A15,XREAL_1:29;
            then r3<G*(1,1)`1 by A13,XXREAL_0:2;
            then
            u0 in { |[r1,s1]| : r1 < G*(1,1)`1 & s1 < G* (1,1)`2 } by A16;
            then
A17:        u0 in Y by A11,GOBOARD6:18;
            reconsider v0=u0 as Element of REAL 2;
A18:        q0`1=r3 & q0`2=s3 by EUCLID:52;
            sqrt 2/2<1 by Lm1,SQUARE_1:21,XREAL_1:189;
            then
A19:        r*(sqrt 2/2)<r*1 by A8,XREAL_1:68;
            (r/2)^2 + (r/2)^2= 2*((r/2)^2)
              .=(sqrt 2)^2*((r/2)^2) by SQUARE_1:def 2
              .=(r/2*sqrt 2)^2;
            then
A20:        sqrt ((r/2)^2 + (r/2)^2) =r*((sqrt 2)/2) by A8,Lm1,SQUARE_1:22;
A21:        r2-r3 =r/2 & s2-s3=r/2;
            p`1=r2 & p`2=s2 by A12,EUCLID:52;
            then dist(u,u0) = (Pitag_dist 2).(v,v0) & (Pitag_dist 2).(v,v0)<r
            by A18,A21,A19,A20,METRIC_1:def 1,TOPREAL3:7;
            then u0 in Ball(u,r) by METRIC_1:11;
            hence Y /\ G0 <> {}(TOP-REAL 2) by A9,A17,XBOOLE_0:def 4;
          end;
          case
A22:        i=0 & j=width G;
            then p in { |[r2,s2]| : r2 <= G*(1,1)`1 & G*(1,width G)`2 <=s2 }
            by A3,Th25;
            then consider r2,s2 such that
A23:        p=|[r2,s2]| and
A24:        r2 <= G*(1,1)`1 and
A25:        G*(1,width G)`2<=s2;
            set r3=r2-r/2,s3=s2+r/2;
A26:        r*2">0 by A8,XREAL_1:129;
            then s3>s2 by XREAL_1:29;
            then
A27:        s3>G*(1,width G)`2 by A25,XXREAL_0:2;
            reconsider q0=|[r3,s3]| as Point of TOP-REAL 2;
            reconsider u0=q0 as Point of Euclid 2 by EUCLID:22;
            r3<r3+r/2 by A26,XREAL_1:29;
            then r3<G*(1,1)`1 by A24,XXREAL_0:2;
            then u0 in { |[r1,s1]| : r1 < G*(1,1)`1 & G*(1,width G)`2<s1 } by
A27;
            then
A28:        u0 in Y by A22,GOBOARD6:19;
            reconsider v0=u0 as Element of REAL 2;
A29:        q0`1=r3 & q0`2=s3 by EUCLID:52;
            sqrt 2/2<1 by Lm1,SQUARE_1:21,XREAL_1:189;
            then
A30:        r*(sqrt 2/2)<r*1 by A8,XREAL_1:68;
            (r/2)^2 + (r/2)^2= 2*((r/2)^2)
              .=(sqrt 2)^2*((r/2)^2) by SQUARE_1:def 2
              .=(r/2*sqrt 2)^2;
            then
A31:        dist(u,u0) = (Pitag_dist 2).(v,v0) & sqrt ((r2 - r3)^2 + (s2
            - s3)^2)<r by A8,A30,Lm1,METRIC_1:def 1,SQUARE_1:22;
            p`1=r2 & p`2=s2 by A23,EUCLID:52;
            then dist(u,u0) < r by A29,A31,TOPREAL3:7;
            then u0 in Ball(u,r) by METRIC_1:11;
            hence Y /\ G0 <> {}(TOP-REAL 2) by A9,A28,XBOOLE_0:def 4;
          end;
          case
A32:        i=0 & 1<=j & j<width G;
            then
            p in { |[r2,s2]| : r2 <= G*(1,1)`1 & G*(1,j)`2 <=s2 & s2<=G*(
            1,j+1)`2} by A3,Th26;
            then consider r2,s2 such that
A33:        p=|[r2,s2]| and
A34:        r2 <= G*(1,1)`1 and
A35:        G*(1,j)`2 <=s2 and
A36:        s2<=G*(1,j+1)`2;
            now
              per cases by A35,A36,XXREAL_0:1;
              case
A37:            s2=G*(1,j)`2;
A38:            p`1=r2 & p`2=s2 by A33,EUCLID:52;
                sqrt 2/2<1 by Lm1,SQUARE_1:21,XREAL_1:189;
                then
A39:            r*(sqrt 2/2)<r*1 by A8,XREAL_1:68;
                (r/2)^2 + (r/2)^2= 2*((r/2)^2)
                  .=(sqrt 2)^2*((r/2)^2) by SQUARE_1:def 2
                  .=(r/2*sqrt 2)^2;
                then
A40:            sqrt ((r/2)^2 + (r/2)^2) =r*((sqrt 2)/2) by A8,Lm1,SQUARE_1:22;
                set rl=G*(1,j+1)`2 - G*(1,j)`2;
                set rm=min(r,rl);
                set r3=r2-r/2 ,s3=s2+rm/2;
                set q0=|[r3,s3]|;
A41:            q0`1=r3 & q0`2=s3 by EUCLID:52;
                reconsider u0=q0 as Point of Euclid 2 by EUCLID:22;
                reconsider v0=u0 as Element of REAL 2;
A42:            1<=len G by Th34;
                j<j+1 & j+1<=width G by A32,NAT_1:13;
                then G*(1,j)`2<G*(1,j+1)`2 by A32,A42,GOBOARD5:4;
                then
A43:            rl>0 by XREAL_1:50;
                then
A44:            rm>0 by A8,XXREAL_0:21;
                then s3>s2 by XREAL_1:29,139;
                then
A45:            s3>G*(1,j)`2 by A35,XXREAL_0:2;
                rm/2<=r/2 by XREAL_1:72,XXREAL_0:17;
                then (rm/2)^2<=(r/2)^2 by A44,SQUARE_1:15;
                then
A46:            (r/2)^2+(rm/2)^2<=(r/2)^2+(r/2)^2 by XREAL_1:7;
                0<=(rm/2)^2 & 0<=(r/2)^2 by XREAL_1:63;
                then
sqrt((r/2)^2+(rm/2)^2)<=sqrt((r/2)^2+(r/2) ^2) by A46,SQUARE_1:26;
                then dist(u,u0) = (Pitag_dist 2).(v,v0) & sqrt ((r2 - r3)^2 +
                (s2 - s3)^2)<r by A39,A40,METRIC_1:def 1,XXREAL_0:2;
                then dist(u,u0) < r by A38,A41,TOPREAL3:7;
                then
A47:            u0 in Ball(u,r) by METRIC_1:11;
                rm/2<=rl/2 by XREAL_1:72,XXREAL_0:17;
                then
A48:            G*(1,j)`2+rm/2<=G*(1,j)`2+rl/2 by XREAL_1:6;
                r*2">0 by A8,XREAL_1:129;
                then r3<r3+r/2 by XREAL_1:29;
                then
A49:            r3<G*(1,1)`1 by A34,XXREAL_0:2;
                G*(1,j)`2+(G*(1,j+1)`2-G*(1,j)`2)/2 <G*(1,j)`2+(G*(1,j+1)
`2-G*(1,j)`2)/2 +(G*(1,j+1)`2-G*(1,j)`2)/2 by A43,XREAL_1:29,139;
                then s3<G*(1,j+1)`2 by A37,A48,XXREAL_0:2;
                then
                u0 in { |[r1,s1]| : r1 < G*(1,1)`1 & G*(1,j)`2<s1 & s1<G*
                (1,j+1)`2} by A49,A45;
                then u0 in Y by A32,GOBOARD6:20;
                hence Y /\ G0 <> {}(TOP-REAL 2) by A9,A47,XBOOLE_0:def 4;
              end;
              case
A50:            G*(1,j)`2 <s2 & s2<G*(1,j+1)`2;
                set r3=r2-r/2,s3=s2;
                reconsider q0=|[r3,s3]| as Point of TOP-REAL 2;
                reconsider u0=q0 as Point of Euclid 2 by EUCLID:22;
                r*2">0 by A8,XREAL_1:129;
                then r3<r3+r/2 by XREAL_1:29;
                then r3<G*(1,1)`1 by A34,XXREAL_0:2;
                then u0 in { |[r1,s1]| : r1 < G*(1,1)`1 & G*(1,j)`2<s1 & s1<G
                *(1,j+1)`2} by A50;
                then
A51:            u0 in Y by A32,GOBOARD6:20;
                reconsider v0=u0 as Element of REAL 2;
A52:            q0`1=r3 & q0`2=s3 by EUCLID:52;
                sqrt 2/2<1 by Lm1,SQUARE_1:21,XREAL_1:189;
                then
A53:            r*(sqrt 2/2)<r*1 by A8,XREAL_1:68;
A54:            (r/2)^2 >=0 by XREAL_1:63;
                then (r/2)^2 + 0 <=(r/2)^2 + (r/2)^2 by XREAL_1:6;
                then
A55:            sqrt((r/2)^2 + 0^2)<=sqrt((r/2)^2 + (r/2)^2) by A54,SQUARE_1:26
;
A56:            p`1=r2 & p`2=s2 by A33,EUCLID:52;
                (r/2)^2 + (r/2)^2= 2*((r/2)^2)
                  .=(sqrt 2)^2*((r/2)^2) by SQUARE_1:def 2
                  .=(r/2*sqrt 2)^2;
                then sqrt ((r/2)^2 + (r/2)^2) =r*((sqrt 2)/2) by A8,Lm1,
SQUARE_1:22;
                then dist(u,u0) = (Pitag_dist 2).(v,v0) & sqrt((r2 - r3)^2 +
                (s2 - s3)^2)<r by A53,A55,METRIC_1:def 1,XXREAL_0:2;
                then dist(u,u0) < r by A56,A52,TOPREAL3:7;
                then u0 in Ball(u,r) by METRIC_1:11;
                hence Y /\ G0 <> {} by A9,A51,XBOOLE_0:def 4;
              end;
              case
A57:            s2=G*(1,j+1)`2;
A58:            p`1=r2 & p`2=s2 by A33,EUCLID:52;
                sqrt 2/2<1 by Lm1,SQUARE_1:21,XREAL_1:189;
                then
A59:            r*(sqrt 2/2)<r*1 by A8,XREAL_1:68;
                (r/2)^2 + (r/2)^2= 2*((r/2)^2)
                  .=(sqrt 2)^2*((r/2)^2) by SQUARE_1:def 2
                  .=(r/2*sqrt 2)^2;
                then
A60:            sqrt ((r/2)^2 + (r/2)^2) =r*((sqrt 2)/2) by A8,Lm1,SQUARE_1:22;
                set rl=G*(1,j+1)`2 - G*(1,j)`2;
                set rm=min(r,rl);
                set r3=r2-r/2,s3=s2-rm/2;
                reconsider q0=|[r3,s3]| as Point of TOP-REAL 2;
A61:            q0`1=r3 & q0`2=s3 by EUCLID:52;
                reconsider u0=q0 as Point of Euclid 2 by EUCLID:22;
                reconsider v0=u0 as Element of REAL 2;
A62:            1<=len G by Th34;
                j<j+1 & j+1<=width G by A32,NAT_1:13;
                then G*(1,j)`2<G*(1,j+1)`2 by A32,A62,GOBOARD5:4;
                then
A63:            rl>0 by XREAL_1:50;
                then
A64:            rm>0 by A8,XXREAL_0:21;
                then s3<s3+rm/2 by XREAL_1:29,139;
                then
A65:            s3<G*(1,j+1)`2 by A36,XXREAL_0:2;
                rm/2<=r/2 by XREAL_1:72,XXREAL_0:17;
                then (rm/2)^2<=(r/2)^2 by A64,SQUARE_1:15;
                then
A66:            (r/2)^2+(rm/2)^2<=(r/2)^2+(r/2)^2 by XREAL_1:7;
                0<=(rm/2)^2 & 0<=(r/2)^2 by XREAL_1:63;
                then sqrt((r/2)^2+(rm/2)^2)<=sqrt((r/2)^2+(r/2) ^2) by A66,
SQUARE_1:26;
                then dist(u,u0) = (Pitag_dist 2).(v,v0) & sqrt ((r2 - r3)^2 +
                (s2 - s3)^2)<r by A59,A60,METRIC_1:def 1,XXREAL_0:2;
                then dist(u,u0) < r by A58,A61,TOPREAL3:7;
                then
A67:            u0 in Ball(u,r) by METRIC_1:11;
                rm/2<=rl/2 by XREAL_1:72,XXREAL_0:17;
                then
A68:            G*(1,j+1)`2-rm/2>=G*(1,j+1)`2-rl/2 by XREAL_1:10;
                r*2">0 by A8,XREAL_1:129;
                then r3<r3+r/2 by XREAL_1:29;
                then
A69:            r3<G*(1,1)`1 by A34,XXREAL_0:2;
                G*(1,j+1)`2-(G*(1,j+1)`2-G*(1,j)`2)/2 >G*(1,j+1)`2-(G*(1
,j+1)`2-G*(1,j)`2)/2 -(G*(1,j+1)`2-G*(1,j)`2)/2 by A63,XREAL_1:44,139;
                then s3>G*(1,j)`2 by A57,A68,XXREAL_0:2;
                then u0 in { |[r1,s1]| : r1 < G*(1,1)`1 & G*(1,j)`2<s1 & s1<G
                *(1,j+1)`2} by A69,A65;
                then u0 in Y by A32,GOBOARD6:20;
                hence Y /\ G0 <> {} by A9,A67,XBOOLE_0:def 4;
              end;
            end;
            hence Y /\ G0 <> {}(TOP-REAL 2);
          end;
          case
A70:        i=len G & j=0;
            then p in { |[r2,s2]| : r2 >= G*(len G,1)`1 & G*(1,1)`2 >=s2 } by
A3,Th27;
            then consider r2,s2 such that
A71:        p=|[r2,s2]| and
A72:        r2 >= G*(len G,1)`1 and
A73:        G*(1,1)`2>=s2;
            set r3=r2+r/2,s3=s2-r/2;
A74:        r*2">0 by A8,XREAL_1:129;
            then r3>r2 by XREAL_1:29;
            then
A75:        r3>G*(len G,1)`1 by A72,XXREAL_0:2;
            reconsider q0=|[r3,s3]| as Point of TOP-REAL 2;
            reconsider u0=q0 as Point of Euclid 2 by EUCLID:22;
            s3<s3+r/2 by A74,XREAL_1:29;
            then s3<G*(1,1)`2 by A73,XXREAL_0:2;
            then u0 in { |[r1,s1]| : r1 > G*(len G,1)`1 & G*(1,1)`2>s1 } by A75
;
            then
A76:        u0 in Y by A70,GOBOARD6:21;
            reconsider v0=u0 as Element of REAL 2;
A77:        q0`1=r3 & q0`2=s3 by EUCLID:52;
            sqrt 2/2<1 by Lm1,SQUARE_1:21,XREAL_1:189;
            then
A78:        r*(sqrt 2/2)<r*1 by A8,XREAL_1:68;
            (r/2)^2 + (r/2)^2= 2*((r/2)^2)
              .=(sqrt 2)^2*((r/2)^2) by SQUARE_1:def 2
              .=(r/2*sqrt 2)^2;
            then
A79:        dist(u,u0) = (Pitag_dist 2).(v,v0) & sqrt ((r2 - r3)^2 + (s2
            - s3)^2)<r by A8,A78,Lm1,METRIC_1:def 1,SQUARE_1:22;
            p`1=r2 & p`2=s2 by A71,EUCLID:52;
            then dist(u,u0) < r by A77,A79,TOPREAL3:7;
            then u0 in Ball(u,r) by METRIC_1:11;
            hence Y /\ G0 <> {} by A9,A76,XBOOLE_0:def 4;
          end;
          case
A80:        i=len G & j=width G;
            sqrt 2/2<1 by Lm1,SQUARE_1:21,XREAL_1:189;
            then
A81:        r*(sqrt 2/2)<r*1 by A8,XREAL_1:68;
            (r/2)^2 + (r/2)^2= 2*((r/2)^2)
              .=(sqrt 2)^2*((r/2)^2) by SQUARE_1:def 2
              .=(r/2*sqrt 2)^2;
            then
A82:        sqrt ((r/2)^2 + (r/2)^2) =r*((sqrt 2)/2) by A8,Lm1,SQUARE_1:22;
            p in { |[r2,s2]| : G*(len G,1)`1<=r2 & G*(1,width G)`2<= s2
            } by A3,A80,Th28;
            then consider r2,s2 such that
A83:        p=|[r2,s2]| and
A84:        G*(len G,1)`1<=r2 and
A85:        G*(1,width G)`2<=s2;
            set r3=r2+r/2,s3=s2+r/2;
A86:        r*2">0 by A8,XREAL_1:129;
            then s2<s2+r/2 by XREAL_1:29;
            then
A87:        s3>G*(1,width G)`2 by A85,XXREAL_0:2;
            reconsider q0=|[r3,s3]| as Point of TOP-REAL 2;
            reconsider u0=q0 as Point of Euclid 2 by EUCLID:22;
            r2<r2+r/2 by A86,XREAL_1:29;
            then r3>G*(len G,1)`1 by A84,XXREAL_0:2;
            then u0 in { |[r1,s1]| : r1 > G*(len G,1)`1 & s1 > G*(1,width G)
            `2 } by A87;
            then
A88:        u0 in Y by A80,GOBOARD6:22;
            reconsider v0=u0 as Element of REAL 2;
A89:        q0`1=r3 & q0`2=s3 by EUCLID:52;
A90:        (-r/2)^2=(r/2)^2 & dist(u,u0) = (Pitag_dist 2).(v,v0) by
METRIC_1:def 1;
A91:        r2-r3=-r/2 & s2-s3=-r/2;
            p`1=r2 & p`2=s2 by A83,EUCLID:52;
            then dist(u,u0) < r by A89,A91,A90,A81,A82,TOPREAL3:7;
            then u0 in Ball(u,r) by METRIC_1:11;
            hence Y /\ G0 <> {} by A9,A88,XBOOLE_0:def 4;
          end;
          case
A92:        i=len G & 1<=j & j<width G;
            then p in { |[r2,s2]| : r2 >= G*(len G,1)`1 & G*(1,j)`2 <=s2 & s2
            <=G*(1,j+1)`2} by A3,Th29;
            then consider r2,s2 such that
A93:        p=|[r2,s2]| and
A94:        r2 >= G*(len G,1)`1 and
A95:        G*(1,j)`2 <=s2 and
A96:        s2<=G*(1,j+1)`2;
            now
              per cases by A95,A96,XXREAL_0:1;
              case
A97:            s2=G*(1,j)`2;
A98:            p`1=r2 & p`2=s2 by A93,EUCLID:52;
                sqrt 2/2<1 by Lm1,SQUARE_1:21,XREAL_1:189;
                then
A99:            r*(sqrt 2/2)<r*1 by A8,XREAL_1:68;
                (r/2)^2 + (r/2)^2= 2*((r/2)^2)
                  .=(sqrt 2)^2*((r/2)^2) by SQUARE_1:def 2
                  .=(r/2*sqrt 2)^2;
                then
A100:           sqrt ((r/2)^2 + (r/2)^2) =r*((sqrt 2)/2) by A8,Lm1,SQUARE_1:22;
                set rl=G*(1,j+1)`2 - G*(1,j)`2;
                set rm=min(r,rl);
                set r3=r2+r/2 ,s3=s2+rm/2;
                reconsider q0=|[r3,s3]| as Point of TOP-REAL 2;
A101:           q0`1=r3 & q0`2=s3 by EUCLID:52;
                reconsider u0=q0 as Point of Euclid 2 by EUCLID:22;
                reconsider v0=u0 as Element of REAL 2;
A102:           1<=len G by Th34;
                j<j+1 & j+1<=width G by A92,NAT_1:13;
                then G*(1,j)`2<G*(1,j+1)`2 by A92,A102,GOBOARD5:4;
                then
A103:           rl>0 by XREAL_1:50;
                then
A104:           rm>0 by A8,XXREAL_0:21;
                then s3>s2 by XREAL_1:29,139;
                then
A105:           s3>G*(1,j)`2 by A95,XXREAL_0:2;
                rm/2<=r/2 by XREAL_1:72,XXREAL_0:17;
                then (rm/2)^2<=(r/2)^2 by A104,SQUARE_1:15;
                then
A106:           (r/2)^2+(rm/2)^2<=(r/2)^2+(r/2)^2 by XREAL_1:7;
                0<=(rm/2)^2 & 0<=(r/2)^2 by XREAL_1:63;
                then sqrt((r/2)^2+(rm/2)^2)<=sqrt((r/2)^2+(r/2) ^2 ) by A106,
SQUARE_1:26;
                then dist(u,u0) = (Pitag_dist 2).(v,v0) & sqrt ((r2 - r3)^2 +
                (s2 - s3)^2)<r by A99,A100,METRIC_1:def 1,XXREAL_0:2;
                then dist(u,u0) < r by A98,A101,TOPREAL3:7;
                then
A107:           u0 in Ball(u,r) by METRIC_1:11;
                rm/2<=rl/2 by XREAL_1:72,XXREAL_0:17;
                then
A108:           G*(1,j)`2+rm/2<=G*(1,j)`2+rl/2 by XREAL_1:6;
                r*2">0 by A8,XREAL_1:129;
                then r2<r2+r/2 by XREAL_1:29;
                then
A109:           r3>G*(len G,1)`1 by A94,XXREAL_0:2;
                G*(1,j)`2+(G*(1,j+1)`2-G*(1,j)`2)/2 <G*(1,j)`2+(G*(1,j+1
)`2-G*(1,j)`2)/2 +(G*(1,j+1)`2-G*(1,j)`2)/2 by A103,XREAL_1:29,139;
                then s3<G*(1,j+1)`2 by A97,A108,XXREAL_0:2;
                then u0 in { |[r1,s1]| : r1 > G*(len G,1)`1 & G*(1,j)`2<s1 &
                s1<G*(1,j+1)`2} by A109,A105;
                then u0 in Y by A92,GOBOARD6:23;
                hence Y /\ G0 <> {} by A9,A107,XBOOLE_0:def 4;
              end;
              case
A110:           G*(1,j)`2 <s2 & s2<G*(1,j+1)`2;
                set r3=r2+r/2,s3=s2;
                reconsider q0=|[r3,s3]| as Point of TOP-REAL 2;
                reconsider u0=q0 as Point of Euclid 2 by EUCLID:22;
                r*2">0 by A8,XREAL_1:129;
                then r2<r2+r/2 by XREAL_1:29;
                then r3>G*(len G,1)`1 by A94,XXREAL_0:2;
                then u0 in { |[r1,s1]| : r1 > G*(len G,1)`1 & G*(1,j)`2<s1 &
                s1<G*(1,j+1)`2} by A110;
                then
A111:           u0 in Y by A92,GOBOARD6:23;
                reconsider v0=u0 as Element of REAL 2;
A112:           q0`1=r3 & q0`2=s3 by EUCLID:52;
                sqrt 2/2<1 by Lm1,SQUARE_1:21,XREAL_1:189;
                then
A113:           r*(sqrt 2/2)<r*1 by A8,XREAL_1:68;
A114:           (r/2)^2 >=0 by XREAL_1:63;
                then (r/2)^2 + 0 <=(r/2)^2 + (r/2)^2 by XREAL_1:6;
                then
A115:           sqrt((r/2)^2 + 0^2)<=sqrt((r/2)^2 + (r/2)^2) by A114,
SQUARE_1:26;
A116:           p`1=r2 & p`2=s2 by A93,EUCLID:52;
                (r/2)^2 + (r/2)^2= 2*((r/2)^2)
                  .=(sqrt 2)^2*((r/2)^2) by SQUARE_1:def 2
                  .=(r/2*sqrt 2)^2;
                then sqrt ((r/2)^2 + (r/2)^2) =r*((sqrt 2)/2) by A8,Lm1,
SQUARE_1:22;
                then dist(u,u0) = (Pitag_dist 2).(v,v0) & sqrt((r2 - r3)^2 +
                (s2 - s3)^2)<r by A113,A115,METRIC_1:def 1,XXREAL_0:2;
                then dist(u,u0) < r by A116,A112,TOPREAL3:7;
                then u0 in Ball(u,r) by METRIC_1:11;
                hence Y /\ G0 <> {} by A9,A111,XBOOLE_0:def 4;
              end;
              case
A117:           s2=G*(1,j+1)`2;
A118:           p`1=r2 & p`2=s2 by A93,EUCLID:52;
                sqrt 2/2<1 by Lm1,SQUARE_1:21,XREAL_1:189;
                then
A119:           r*(sqrt 2/2)<r*1 by A8,XREAL_1:68;
                (r/2)^2 + (r/2)^2= 2*((r/2)^2)
                  .=(sqrt 2)^2*((r/2)^2) by SQUARE_1:def 2
                  .=(r/2*sqrt 2)^2;
                then
A120:           sqrt ((r/2)^2 + (r/2)^2) =r*((sqrt 2)/2) by A8,Lm1,SQUARE_1:22;
                set rl=G*(1,j+1)`2 - G*(1,j)`2;
                set rm=min(r,rl);
                set r3=r2+r/2,s3=s2-rm/2;
                reconsider q0=|[r3,s3]| as Point of TOP-REAL 2;
A121:           q0`1=r3 & q0`2=s3 by EUCLID:52;
                reconsider u0=q0 as Point of Euclid 2 by EUCLID:22;
                reconsider v0=u0 as Element of REAL 2;
A122:           1<=len G by Th34;
                j<j+1 & j+1<=width G by A92,NAT_1:13;
                then G*(1,j)`2<G*(1,j+1)`2 by A92,A122,GOBOARD5:4;
                then
A123:           rl>0 by XREAL_1:50;
                then
A124:           rm>0 by A8,XXREAL_0:21;
                then s3<s3+rm/2 by XREAL_1:29,139;
                then
A125:           s3<G*(1,j+1)`2 by A96,XXREAL_0:2;
                rm/2<=r/2 by XREAL_1:72,XXREAL_0:17;
                then (rm/2)^2<=(r/2)^2 by A124,SQUARE_1:15;
                then
A126:           (r/2)^2+(rm/2)^2<=(r/2)^2+(r/2)^2 by XREAL_1:7;
                0<=(rm/2)^2 & 0<=(r/2)^2 by XREAL_1:63;
                then sqrt((r/2)^2+(rm/2)^2)<=sqrt((r/2)^2+(r/2) ^2 ) by A126,
SQUARE_1:26;
                then dist(u,u0) = (Pitag_dist 2).(v,v0) & sqrt ((r2 - r3)^2 +
                (s2 - s3)^2)<r by A119,A120,METRIC_1:def 1,XXREAL_0:2;
                then dist(u,u0) < r by A118,A121,TOPREAL3:7;
                then
A127:           u0 in Ball(u,r) by METRIC_1:11;
                rm/2<=rl/2 by XREAL_1:72,XXREAL_0:17;
                then
A128:           G*(1,j+1)`2-rm/2>=G*(1,j+1)`2-rl/2 by XREAL_1:10;
                r*2">0 by A8,XREAL_1:129;
                then r2<r2+r/2 by XREAL_1:29;
                then
A129:           r3>G*(len G,1)`1 by A94,XXREAL_0:2;
                G*(1,j+1)`2-(G*(1,j+1)`2-G*(1,j)`2)/2 >G*(1,j+1)`2-(G*(1
,j+1)`2-G*(1,j)`2)/2 -(G*(1,j+1)`2-G*(1,j)`2)/2 by A123,XREAL_1:44,139;
                then s3>G*(1,j)`2 by A117,A128,XXREAL_0:2;
                then u0 in { |[r1,s1]| : r1 > G*(len G,1)`1 & G*(1,j)`2<s1 &
                s1<G*(1,j+1)`2} by A129,A125;
                then u0 in Y by A92,GOBOARD6:23;
                hence Y /\ G0 <> {}(TOP-REAL 2) by A9,A127,XBOOLE_0:def 4;
              end;
            end;
            hence Y /\ G0 <> {}(TOP-REAL 2);
          end;
          case
A130:       1<=i & i<len G & j=0;
            then p in { |[r2,s2]| : G*(i,1)`1 <=r2 & r2<=G*(i+1,1)`1 & s2 <=
            G*(1,1)`2 } by A3,Th30;
            then consider r2,s2 such that
A131:       p=|[r2,s2]| and
A132:       G*(i,1)`1 <=r2 and
A133:       r2<=G*(i+1,1)`1 and
A134:       s2 <= G*(1,1)`2;
            now
              per cases by A132,A133,XXREAL_0:1;
              case
A135:           r2=G*(i,1)`1;
                (r/2)^2 + (r/2)^2= 2*((r/2)^2)
                  .=(sqrt 2)^2*((r/2)^2) by SQUARE_1:def 2
                  .=(r/2*sqrt 2)^2;
                then
A136:           sqrt ((r/2)^2 + (r/2)^2) =r*((sqrt 2)/2) by A8,Lm1,SQUARE_1:22;
                set sl=G*(i+1,1)`1 - G*(i,1)`1;
                set sm=min(r,sl);
                set s3=s2-r/2,r3=r2+sm/2;
                reconsider q0=|[r3,s3]| as Point of TOP-REAL 2;
A137:           q0`1=r3 & q0`2=s3 by EUCLID:52;
                reconsider u0=q0 as Point of Euclid 2 by EUCLID:22;
                reconsider v0=u0 as Element of REAL 2;
A138:           1<=width G by Th34;
                i<i+1 & i+1<=len G by A130,NAT_1:13;
                then G*(i,1)`1<G*(i+1,1)`1 by A130,A138,GOBOARD5:3;
                then
A139:           sl>0 by XREAL_1:50;
                then
A140:           sm>0 by A8,XXREAL_0:21;
                then r3>r2 by XREAL_1:29,139;
                then
A141:           r3>G*(i,1)`1 by A132,XXREAL_0:2;
                sqrt 2/2<1 by Lm1,SQUARE_1:21,XREAL_1:189;
                then
A142:           r*(sqrt 2/2)<r*1 by A8,XREAL_1:68;
                sm/2<=r/2 by XREAL_1:72,XXREAL_0:17;
                then (sm/2)^2<=(r/2)^2 by A140,SQUARE_1:15;
                then
A143:           (r/2)^2+(sm/2)^2<=(r/2)^2+(r/2)^2 by XREAL_1:7;
                0<=(sm/2)^2 & 0<=(r/2)^2 by XREAL_1:63;
                then
A144:           sqrt((r/2)^2+(sm/2)^2)<=sqrt((r/2)^2+(r/2) ^2 ) by A143,
SQUARE_1:26;
                p`1=r2 & p`2=s2 by A131,EUCLID:52;
                then dist(u,u0) = (Pitag_dist 2).(v,v0) & sqrt ((p`1 - q0`1)
^2 + (p`2 - q0`2)^2)< r by A144,A137,A142,A136,METRIC_1:def 1,XXREAL_0:2;
                then dist(u,u0) < r by TOPREAL3:7;
                then
A145:           u0 in Ball(u,r) by METRIC_1:11;
                sm/2<=sl/2 by XREAL_1:72,XXREAL_0:17;
                then
A146:           G*(i,1)`1+sm/2<=G*(i,1)`1+sl/2 by XREAL_1:6;
                r*2">0 by A8,XREAL_1:129;
                then s3<s3+r/2 by XREAL_1:29;
                then
A147:           s3<G*(1,1)`2 by A134,XXREAL_0:2;
                G*(i,1)`1+(G*(i+1,1)`1-G*(i,1)`1)/2 <G*(i,1)`1+(G*(i+1,1
)`1-G*(i,1)`1)/2 +(G*(i+1,1)`1-G*(i,1)`1)/2 by A139,XREAL_1:29,139;
                then r3<G*(i+1,1)`1 by A135,A146,XXREAL_0:2;
                then u0 in { |[r1,s1]| : G*(i,1)`1<r1 & r1<G*(i+1,1)`1 & s1 <
                G*(1,1)`2} by A147,A141;
                then u0 in Y by A130,GOBOARD6:24;
                hence Y /\ G0 <> {}(TOP-REAL 2) by A9,A145,XBOOLE_0:def 4;
              end;
              case
A148:           G*(i,1)`1 <r2 & r2<G*(i+1,1)`1;
                set s3=s2-r/2,r3=r2;
                reconsider q0=|[r3,s3]| as Point of TOP-REAL 2;
                reconsider u0=q0 as Point of Euclid 2 by EUCLID:22;
                r*2">0 by A8,XREAL_1:129;
                then s3<s3+r/2 by XREAL_1:29;
                then s3<G*(1,1)`2 by A134,XXREAL_0:2;
                then u0 in { |[r1,s1]| : G*(i,1)`1<r1 & r1<G*(i+1,1)`1 & s1 <
                G*(1,1)`2 } by A148;
                then
A149:           u0 in Y by A130,GOBOARD6:24;
                reconsider v0=u0 as Element of REAL 2;
A150:           q0`1=r3 & q0`2=s3 by EUCLID:52;
                sqrt 2/2<1 by Lm1,SQUARE_1:21,XREAL_1:189;
                then
A151:           r*(sqrt 2/2)<r*1 by A8,XREAL_1:68;
A152:           (r/2)^2 >=0 by XREAL_1:63;
                then 0^2 + (r/2)^2 <=(r/2)^2 + (r/2)^2 by XREAL_1:6;
                then
A153:           sqrt(0^2 + (r/2)^2 )<=sqrt((r/2)^2 + (r/2)^2) by A152,
SQUARE_1:26;
A154:           p`1=r2 & p`2=s2 by A131,EUCLID:52;
                (r/2)^2 + (r/2)^2= 2*((r/2)^2)
                  .=(sqrt 2)^2*((r/2)^2) by SQUARE_1:def 2
                  .=(r/2*sqrt 2)^2;
                then sqrt ((r/2)^2 + (r/2)^2) =r*((sqrt 2)/2) by A8,Lm1,
SQUARE_1:22;
                then dist(u,u0) = (Pitag_dist 2).(v,v0) & sqrt((r2 - r3)^2 +
                (s2 - s3)^2 )<r by A151,A153,METRIC_1:def 1,XXREAL_0:2;
                then dist(u,u0) < r by A154,A150,TOPREAL3:7;
                then u0 in Ball(u,r) by METRIC_1:11;
                hence Y /\ G0 <> {}(TOP-REAL 2) by A9,A149,XBOOLE_0:def 4;
              end;
              case
A155:           r2=G*(i+1,1)`1;
                (r/2)^2 + (r/2)^2= 2*((r/2)^2)
                  .=(sqrt 2)^2*((r/2)^2) by SQUARE_1:def 2
                  .=(r/2*sqrt 2)^2;
                then
A156:           sqrt ((r/2)^2 + (r/2)^2) =r*((sqrt 2)/2) by A8,Lm1,SQUARE_1:22;
                set sl=G*(i+1,1)`1 - G*(i,1)`1;
                set sm=min(r,sl);
                set s3=s2-r/2,r3=r2-sm/2;
                reconsider q0=|[r3,s3]| as Point of TOP-REAL 2;
A157:           q0`1=r3 & q0`2=s3 by EUCLID:52;
                reconsider u0=q0 as Point of Euclid 2 by EUCLID:22;
                reconsider v0=u0 as Element of REAL 2;
A158:           1<=width G by Th34;
                i<i+1 & i+1<=len G by A130,NAT_1:13;
                then G*(i,1)`1<G*(i+1,1)`1 by A130,A158,GOBOARD5:3;
                then
A159:           sl>0 by XREAL_1:50;
                then
A160:           sm>0 by A8,XXREAL_0:21;
                then r3<r3+sm/2 by XREAL_1:29,139;
                then
A161:           r3<G*(i+1,1)`1 by A133,XXREAL_0:2;
                sqrt 2/2<1 by Lm1,SQUARE_1:21,XREAL_1:189;
                then
A162:           r*(sqrt 2/2)<r*1 by A8,XREAL_1:68;
                sm/2<=r/2 by XREAL_1:72,XXREAL_0:17;
                then (sm/2)^2<=(r/2)^2 by A160,SQUARE_1:15;
                then
A163:           (r/2)^2+(sm/2)^2<=(r/2)^2+(r/2)^2 by XREAL_1:7;
                0<=(sm/2)^2 & 0<=(r/2)^2 by XREAL_1:63;
                then
A164:           sqrt((r/2)^2+(sm/2)^2)<=sqrt((r/2)^2+(r/2) ^2 ) by A163,
SQUARE_1:26;
                p`1=r2 & p`2=s2 by A131,EUCLID:52;
                then dist(u,u0) = (Pitag_dist 2).(v,v0) & sqrt ((p`1 - q0`1)
^2 + (p`2 - q0`2)^2)< r by A164,A157,A162,A156,METRIC_1:def 1,XXREAL_0:2;
                then dist(u,u0) < r by TOPREAL3:7;
                then
A165:           u0 in Ball(u,r) by METRIC_1:11;
                sm/2<=sl/2 by XREAL_1:72,XXREAL_0:17;
                then
A166:           G*(i+1,1)`1-sm/2>=G*(i+1,1)`1-sl/2 by XREAL_1:10;
                r*2">0 by A8,XREAL_1:129;
                then s3<s3+r/2 by XREAL_1:29;
                then
A167:           s3<G*(1,1)`2 by A134,XXREAL_0:2;
                G*(i+1,1)`1-(G*(i+1,1)`1-G*(i,1)`1)/2 >G*(i+1,1)`1-(G*(i
+1,1)`1-G*(i,1)`1)/2 -(G*(i+1,1)`1-G*(i,1)`1)/2 by A159,XREAL_1:44,139;
                then r3>G*(i,1)`1 by A155,A166,XXREAL_0:2;
                then u0 in { |[r1,s1]| : G*(i,1)`1<r1 & r1<G*(i+1,1)`1 & s1 <
                G*(1,1)`2 } by A167,A161;
                then u0 in Y by A130,GOBOARD6:24;
                hence Y /\ G0 <> {}(TOP-REAL 2) by A9,A165,XBOOLE_0:def 4;
              end;
            end;
            hence Y /\ G0 <> {}(TOP-REAL 2);
          end;
          case
A168:       1<=i & i<len G & j=width G;
            then p in { |[r2,s2]| : G*(i,1)`1 <=r2 & r2<=G*(i+1,1)`1 & s2>=G*
            (1,width G)`2} by A3,Th31;
            then consider r2,s2 such that
A169:       p=|[r2,s2]| and
A170:       G*(i,1)`1 <=r2 and
A171:       r2<=G*(i+1,1)`1 and
A172:       s2>=G*(1,width G)`2;
            now
              per cases by A170,A171,XXREAL_0:1;
              case
A173:           r2=G*(i,1)`1;
A174:           p`1=r2 & p`2=s2 by A169,EUCLID:52;
                sqrt 2/2<1 by Lm1,SQUARE_1:21,XREAL_1:189;
                then
A175:           r*(sqrt 2/2)<r*1 by A8,XREAL_1:68;
                (r/2)^2 + (r/2)^2= 2*((r/2)^2)
                  .=(sqrt 2)^2*((r/2)^2) by SQUARE_1:def 2
                  .=(r/2*sqrt 2)^2;
                then
A176:           sqrt ((r/2)^2 + (r/2)^2) =r*((sqrt 2)/2) by A8,Lm1,SQUARE_1:22;
                set rl=G*(i+1,1)`1 - G*(i,1)`1;
                set rm=min(r,rl);
                set s3=s2+r/2,r3=r2+rm/2;
                reconsider q0=|[r3,s3]| as Point of TOP-REAL 2;
A177:           q0`1=r3 & q0`2=s3 by EUCLID:52;
                reconsider u0=q0 as Point of Euclid 2 by EUCLID:22;
                reconsider v0=u0 as Element of REAL 2;
A178:           1<=width G by Th34;
                i<i+1 & i+1<=len G by A168,NAT_1:13;
                then G*(i,1)`1<G*(i+1,1)`1 by A168,A178,GOBOARD5:3;
                then
A179:           rl>0 by XREAL_1:50;
                then
A180:           rm>0 by A8,XXREAL_0:21;
                then r3>r2 by XREAL_1:29,139;
                then
A181:           r3>G*(i,1)`1 by A170,XXREAL_0:2;
                rm/2<=r/2 by XREAL_1:72,XXREAL_0:17;
                then (rm/2)^2<=(r/2)^2 by A180,SQUARE_1:15;
                then
A182:           (r/2)^2+(rm/2)^2<=(r/2)^2+(r/2)^2 by XREAL_1:7;
                0<=(rm/2)^2 & 0<=(r/2)^2 by XREAL_1:63;
                then sqrt((r/2)^2+(rm/2)^2)<=sqrt((r/2)^2+(r/2) ^2 ) by A182,
SQUARE_1:26;
                then dist(u,u0) = (Pitag_dist 2).(v,v0) & sqrt ((r2 - r3)^2 +
                (s2 - s3)^2)<r by A175,A176,METRIC_1:def 1,XXREAL_0:2;
                then dist(u,u0) < r by A174,A177,TOPREAL3:7;
                then
A183:           u0 in Ball(u,r) by METRIC_1:11;
                rm/2<=rl/2 by XREAL_1:72,XXREAL_0:17;
                then
A184:           G*(i,1)`1+rm/2<=G*(i,1)`1+rl/2 by XREAL_1:6;
                r*2">0 by A8,XREAL_1:129;
                then s2<s2+r/2 by XREAL_1:29;
                then
A185:           s3>G*(1,width G)`2 by A172,XXREAL_0:2;
                G*(i,1)`1+(G*(i+1,1)`1-G*(i,1)`1)/2 <G*(i,1)`1+(G*(i+1,1
)`1-G*(i,1)`1)/2 +(G*(i+1,1)`1-G*(i,1)`1)/2 by A179,XREAL_1:29,139;
                then r3<G*(i+1,1)`1 by A173,A184,XXREAL_0:2;
                then u0 in { |[r1,s1]| : G*(i,1)`1<r1 & r1<G*(i+1,1)`1 & s1 >
                G*(1,width G)`2 } by A185,A181;
                then u0 in Y by A168,GOBOARD6:25;
                hence Y /\ G0 <> {}(TOP-REAL 2) by A9,A183,XBOOLE_0:def 4;
              end;
              case
A186:           G*(i,1)`1 <r2 & r2<G*(i+1,1)`1;
                set s3=s2+r/2,r3=r2;
                reconsider q0=|[r3,s3]| as Point of TOP-REAL 2;
                reconsider u0=q0 as Point of Euclid 2 by EUCLID:22;
                r*2">0 by A8,XREAL_1:129;
                then s2<s3 by XREAL_1:29;
                then s3>G*(1,width G)`2 by A172,XXREAL_0:2;
                then u0 in { |[r1,s1]| : G*(i,1)`1<r1 & r1<G*(i+1,1)`1 & s1 >
                G*(1,width G)`2 } by A186;
                then
A187:           u0 in Y by A168,GOBOARD6:25;
                reconsider v0=u0 as Element of REAL 2;
A188:           q0`1=r3 & q0`2=s3 by EUCLID:52;
                sqrt 2/2<1 by Lm1,SQUARE_1:21,XREAL_1:189;
                then
A189:           r*(sqrt 2/2)<r*1 by A8,XREAL_1:68;
A190:           (r/2)^2 >=0 by XREAL_1:63;
                then (r/2)^2 + 0^2 <=(r/2)^2 + (r/2)^2 by XREAL_1:6;
                then
A191:           sqrt((r/2)^2 + 0^2)<=sqrt((r/2)^2 + (r/2)^2) by A190,
SQUARE_1:26;
A192:           p`1=r2 & p`2=s2 by A169,EUCLID:52;
                (r/2)^2 + (r/2)^2= 2*((r/2)^2)
                  .=(sqrt 2)^2*((r/2)^2) by SQUARE_1:def 2
                  .=(r/2*sqrt 2)^2;
                then sqrt ((r/2)^2 + (r/2)^2) =r*((sqrt 2)/2) by A8,Lm1,
SQUARE_1:22;
                then dist(u,u0) = (Pitag_dist 2).(v,v0) & sqrt((r2 - r3)^2 +
                (s2 - s3)^2)<r by A189,A191,METRIC_1:def 1,XXREAL_0:2;
                then dist(u,u0) < r by A192,A188,TOPREAL3:7;
                then u0 in Ball(u,r) by METRIC_1:11;
                hence Y /\ G0 <> {}(TOP-REAL 2) by A9,A187,XBOOLE_0:def 4;
              end;
              case
A193:           r2=G*(i+1,1)`1;
A194:           p`1=r2 & p`2=s2 by A169,EUCLID:52;
                sqrt 2/2<1 by Lm1,SQUARE_1:21,XREAL_1:189;
                then
A195:           r*(sqrt 2/2)<r*1 by A8,XREAL_1:68;
                (r/2)^2 + (r/2)^2= 2*((r/2)^2)
                  .=(sqrt 2)^2*((r/2)^2) by SQUARE_1:def 2
                  .=(r/2*sqrt 2)^2;
                then
A196:           sqrt ((r/2)^2 + (r/2)^2) =r*((sqrt 2)/2) by A8,Lm1,SQUARE_1:22;
                set rl=G*(i+1,1)`1 - G*(i,1)`1;
                set rm=min(r,rl);
                set s3=s2+r/2,r3=r2-rm/2;
                reconsider q0=|[r3,s3]| as Point of TOP-REAL 2;
A197:           q0`1=r3 & q0`2=s3 by EUCLID:52;
                reconsider u0=q0 as Point of Euclid 2 by EUCLID:22;
                reconsider v0=u0 as Element of REAL 2;
A198:           1<=width G by Th34;
                i<i+1 & i+1<=len G by A168,NAT_1:13;
                then G*(i,1)`1<G*(i+1,1)`1 by A168,A198,GOBOARD5:3;
                then
A199:           rl>0 by XREAL_1:50;
                then
A200:           rm>0 by A8,XXREAL_0:21;
                then r3<r3+rm/2 by XREAL_1:29,139;
                then
A201:           r3<G*(i+1,1)`1 by A171,XXREAL_0:2;
                rm/2<=r/2 by XREAL_1:72,XXREAL_0:17;
                then (rm/2)^2<=(r/2)^2 by A200,SQUARE_1:15;
                then
A202:           (r/2)^2+(rm/2)^2<=(r/2)^2+(r/2)^2 by XREAL_1:7;
                0<=(rm/2)^2 & 0<=(r/2)^2 by XREAL_1:63;
                then sqrt((r/2)^2+(rm/2)^2)<=sqrt((r/2)^2+(r/2) ^2 ) by A202,
SQUARE_1:26;
                then dist(u,u0) = (Pitag_dist 2).(v,v0) & sqrt ((r2 - r3)^2 +
                (s2 - s3)^2)<r by A195,A196,METRIC_1:def 1,XXREAL_0:2;
                then dist(u,u0) < r by A194,A197,TOPREAL3:7;
                then
A203:           u0 in Ball(u,r) by METRIC_1:11;
                rm/2<=rl/2 by XREAL_1:72,XXREAL_0:17;
                then
A204:           G*(i+1,1)`1-rm/2>=G*(i+1,1)`1-rl/2 by XREAL_1:10;
                r*2">0 by A8,XREAL_1:129;
                then s3>s2 by XREAL_1:29;
                then
A205:           s3>G*(1,width G)`2 by A172,XXREAL_0:2;
                G*(i+1,1)`1-(G*(i+1,1)`1-G*(i,1)`1)/2 >G*(i+1,1)`1-(G*(i
+1,1)`1-G*(i,1)`1)/2 -(G*(i+1,1)`1-G*(i,1)`1)/2 by A199,XREAL_1:44,139;
                then r3>G*(i,1)`1 by A193,A204,XXREAL_0:2;
                then u0 in { |[r1,s1]| : G*(i,1)`1<r1 & r1<G*(i+1,1)`1 & s1 >
                G*(1,width G)`2} by A205,A201;
                then u0 in Y by A168,GOBOARD6:25;
                hence Y /\ G0 <> {}(TOP-REAL 2) by A9,A203,XBOOLE_0:def 4;
              end;
            end;
            hence Y /\ G0 <> {}(TOP-REAL 2);
          end;
          case
A206:       1<=i & i<len G & 1<=j & j<width G;
            then p in { |[r2,s2]| : G*(i,1)`1 <=r2 & r2<=G*(i+1,1)`1 & G*(1,j
            )`2 <=s2 & s2<=G*(1,j+1)`2} by A3,Th32;
            then consider r2,s2 such that
A207:       p=|[r2,s2]| and
A208:       G*(i,1)`1 <=r2 and
A209:       r2<=G*(i+1,1)`1 and
A210:       G*(1,j)`2 <=s2 and
A211:       s2<=G*(1,j+1)`2;
            now
              per cases by A208,A209,A210,A211,XXREAL_0:1;
              case
A212:           r2=G*(i,1)`1 & s2=G*(1,j)`2;
                set rl1=G*(i+1,1)`1 - G*(i,1)`1;
                set rl=G*(1,j+1)`2 - G*(1,j)`2;
                set rm=min(r,rl);
                set rm1=min(r,rl1);
                set r3=r2+rm1/2,s3=s2+rm/2;
A213:           1<=width G by Th34;
                i<i+1 & i+1<=len G by A206,NAT_1:13;
                then G*(i,1)`1<G*(i+1,1)`1 by A206,A213,GOBOARD5:3;
                then
A214:           rl1>0 by XREAL_1:50;
                then
A215:           rm1>0 by A8,XXREAL_0:21;
                then r3>r2 by XREAL_1:29,139;
                then
A216:           r3>G*(i,1)`1 by A208,XXREAL_0:2;
                rm1/2<=rl1/2 by XREAL_1:72,XXREAL_0:17;
                then
A217:           G*(i,1)`1+rm1/2<=G*(i,1)`1+rl1/2 by XREAL_1:6;
                G*(i,1)`1+(G*(i+1,1)`1-G*(i,1)`1)/2 <G*(i,1)`1+(G*(i+1,1
)`1-G*(i,1)`1)/2 +(G*(i+1,1)`1-G*(i,1)`1)/2 by A214,XREAL_1:29,139;
                then
A218:           r3<G*(i+1,1)`1 by A212,A217,XXREAL_0:2;
                rm/2<=rl/2 by XREAL_1:72,XXREAL_0:17;
                then
A219:           G*(1,j)`2+rm/2<=G*(1,j)`2+rl/2 by XREAL_1:6;
                reconsider q0=|[r3,s3]| as Point of TOP-REAL 2;
A220:           q0`1=r3 & q0`2=s3 by EUCLID:52;
                reconsider u0=q0 as Point of Euclid 2 by EUCLID:22;
                reconsider v0=u0 as Element of REAL 2;
A221:           1<=len G by Th34;
                j<j+1 & j+1<=width G by A206,NAT_1:13;
                then G*(1,j)`2<G*(1,j+1)`2 by A206,A221,GOBOARD5:4;
                then
A222:           rl>0 by XREAL_1:50;
                then
A223:           rm>0 by A8,XXREAL_0:21;
                then s3>s2 by XREAL_1:29,139;
                then
A224:           s3>G*(1,j)`2 by A210,XXREAL_0:2;
                rm1/2<=r/2 by XREAL_1:72,XXREAL_0:17;
                then
A225:           (rm1/2)^2<=(r/2)^2 by A215,SQUARE_1:15;
                rm/2<=r/2 by XREAL_1:72,XXREAL_0:17;
                then (rm/2)^2<=(r/2)^2 by A223,SQUARE_1:15;
                then
A226:           (rm1/2)^2+(rm/2)^2<=(r/2)^2+(r/2)^2 by A225,XREAL_1:7;
                sqrt 2/2<1 by Lm1,SQUARE_1:21,XREAL_1:189;
                then
A227:           r*(sqrt 2/2)<r*1 by A8,XREAL_1:68;
                (r/2)^2 + (r/2)^2= 2*((r/2)^2)
                  .=(sqrt 2)^2*((r/2)^2) by SQUARE_1:def 2
                  .=(r/2*sqrt 2)^2;
                then
A228:           sqrt ((r/2)^2 + (r/2)^2) =r*((sqrt 2)/2) by A8,Lm1,SQUARE_1:22;
A229:           p`1=r2 & p`2=s2 by A207,EUCLID:52;
                0<=(rm/2)^2 & 0<=(rm1/2)^2 by XREAL_1:63;
                then sqrt((rm1/2)^2+(rm/2)^2)<=sqrt((r/2)^2+(r/ 2)^2 ) by A226,
SQUARE_1:26;
                then dist(u,u0) = (Pitag_dist 2).(v,v0) & sqrt ((r2 - r3)^2 +
                (s2 - s3)^2)<r by A227,A228,METRIC_1:def 1,XXREAL_0:2;
                then dist(u,u0) < r by A229,A220,TOPREAL3:7;
                then
A230:           u0 in Ball(u,r) by METRIC_1:11;
                G*(1,j)`2+(G*(1,j+1)`2-G*(1,j)`2)/2 <G*(1,j)`2+(G*(1,j+1
)`2-G*(1,j)`2)/2 +(G*(1,j+1)`2-G*(1,j)`2)/2 by A222,XREAL_1:29,139;
                then s3<G*(1,j+1)`2 by A212,A219,XXREAL_0:2;
                then u0 in { |[r1,s1]| : G*(i,1)`1 < r1 & r1 < G*(i+1,1)`1 &
                G*(1,j)`2 < s1 & s1 < G*(1,j+1)`2 } by A224,A216,A218;
                then u0 in Y by A206,GOBOARD6:26;
                hence Y /\ G0 <> {}(TOP-REAL 2) by A9,A230,XBOOLE_0:def 4;
              end;
              case
A231:           r2=G*(i,1)`1 & G*(1,j)`2 <s2 & s2<G*(1,j+1)`2;
                set s3=s2;
                set rl1=G*(i+1,1)`1 - G*(i,1)`1;
                set rm1=min(r,rl1);
                set r3=r2+rm1/2;
                reconsider q0=|[r3,s3]| as Point of TOP-REAL 2;
                reconsider u0=q0 as Point of Euclid 2 by EUCLID:22;
A232:           1<=width G by Th34;
                i<i+1 & i+1<=len G by A206,NAT_1:13;
                then G*(i,1)`1<G*(i+1,1)`1 by A206,A232,GOBOARD5:3;
                then
A233:           rl1>0 by XREAL_1:50;
                then
A234:           rm1>0 by A8,XXREAL_0:21;
                then
A235:           r3>r2 by XREAL_1:29,139;
                rm1/2<=rl1/2 by XREAL_1:72,XXREAL_0:17;
                then
A236:           G*(i,1)`1+rm1/2<=G*(i,1)`1+rl1/2 by XREAL_1:6;
                G*(i,1)`1+(G*(i+1,1)`1-G*(i,1)`1)/2 <G*(i,1)`1+(G*(i+1,1
)`1-G*(i,1)`1)/2 +(G*(i+1,1)`1-G*(i,1)`1)/2 by A233,XREAL_1:29,139;
                then r3<G*(i+1,1)`1 by A231,A236,XXREAL_0:2;
                then u0 in { |[r1,s1]| : G*(i,1)`1 < r1 & r1 < G*(i+1,1)`1 &
                G*(1,j)`2 < s1 & s1 < G*(1,j+1)`2 } by A231,A235;
                then
A237:           u0 in Y by A206,GOBOARD6:26;
                sqrt 2/2<1 by Lm1,SQUARE_1:21,XREAL_1:189;
                then
A238:           r*(sqrt 2/2)<r*1 by A8,XREAL_1:68;
                reconsider v0=u0 as Element of REAL 2;
A239:           0<=(rm1/2)^2 by XREAL_1:63;
                rm1/2<=r/2 by XREAL_1:72,XXREAL_0:17;
                then (rm1/2)^2+0^2<=(r/2)^2+0^2 by A234,SQUARE_1:15;
                then
A240:           sqrt((rm1/2)^2+0^2)<=sqrt((r/2)^2+0^2) by A239,SQUARE_1:26;
A241:           q0`1=r3 & q0`2=s3 by EUCLID:52;
A242:           (r/2)^2 >=0 by XREAL_1:63;
                then (r/2)^2 + 0^2 <=(r/2)^2 + (r/2)^2 by XREAL_1:6;
                then
A243:           sqrt((r/2)^2 + 0^2)<=sqrt((r/2)^2 + (r/2)^2) by A242,
SQUARE_1:26;
                (r/2)^2 + (r/2)^2= 2*((r/2)^2)
                  .=(sqrt 2)^2*((r/2)^2) by SQUARE_1:def 2
                  .=(r/2*sqrt 2)^2;
                then sqrt ((r/2)^2 + (r/2)^2) =r*((sqrt 2)/2) by A8,Lm1,
SQUARE_1:22;
                then sqrt((r/2)^2 + 0^2)<r by A238,A243,XXREAL_0:2;
                then
A244:           dist(u,u0) = (Pitag_dist 2).(v,v0) & sqrt((r2 - r3)^2 +
                (s2 - s3)^2)<r by A240,METRIC_1:def 1,XXREAL_0:2;
                p`1=r2 & p`2=s2 by A207,EUCLID:52;
                then dist(u,u0) < r by A241,A244,TOPREAL3:7;
                then u0 in Ball(u,r) by METRIC_1:11;
                hence Y /\ G0 <> {}(TOP-REAL 2) by A9,A237,XBOOLE_0:def 4;
              end;
              case
A245:           r2=G*(i,1)`1 & s2=G*(1,j+1)`2;
                set rl1=G*(i+1,1)`1 - G*(i,1)`1;
                set rl=G*(1,j+1)`2 - G*(1,j)`2;
                set rm=min(r,rl);
                set rm1=min(r,rl1);
                set r3=r2+rm1/2,s3=s2-rm/2;
A246:           1<=width G by Th34;
                i<i+1 & i+1<=len G by A206,NAT_1:13;
                then G*(i,1)`1<G*(i+1,1)`1 by A206,A246,GOBOARD5:3;
                then
A247:           rl1>0 by XREAL_1:50;
                then
A248:           rm1>0 by A8,XXREAL_0:21;
                then r3>r2 by XREAL_1:29,139;
                then
A249:           r3>G*(i,1)`1 by A208,XXREAL_0:2;
                rm1/2<=rl1/2 by XREAL_1:72,XXREAL_0:17;
                then
A250:           G*(i,1)`1+rm1/2<=G*(i,1)`1+rl1/2 by XREAL_1:6;
                G*(i,1)`1+(G*(i+1,1)`1-G*(i,1)`1)/2 <G*(i,1)`1+(G*(i+1,1
)`1-G*(i,1)`1)/2 +(G*(i+1,1)`1-G*(i,1)`1)/2 by A247,XREAL_1:29,139;
                then
A251:           r3<G*(i+1,1)`1 by A245,A250,XXREAL_0:2;
                rm/2<=rl/2 by XREAL_1:72,XXREAL_0:17;
                then
A252:           G*(1,j+1)`2-rm/2>=G*(1,j+1)`2-rl/2 by XREAL_1:13;
                reconsider q0=|[r3,s3]| as Point of TOP-REAL 2;
A253:           q0`1=r3 & q0`2=s3 by EUCLID:52;
                reconsider u0=q0 as Point of Euclid 2 by EUCLID:22;
                reconsider v0=u0 as Element of REAL 2;
A254:           1<=len G by Th34;
                j<j+1 & j+1<=width G by A206,NAT_1:13;
                then G*(1,j)`2<G*(1,j+1)`2 by A206,A254,GOBOARD5:4;
                then
A255:           rl>0 by XREAL_1:50;
                then
A256:           rm>0 by A8,XXREAL_0:21;
                then s3<s2 by XREAL_1:44,139;
                then
A257:           s3<G*(1,j+1)`2 by A211,XXREAL_0:2;
                rm1/2<=r/2 by XREAL_1:72,XXREAL_0:17;
                then
A258:           (rm1/2)^2<=(r/2)^2 by A248,SQUARE_1:15;
                rm/2<=r/2 by XREAL_1:72,XXREAL_0:17;
                then (rm/2)^2<=(r/2)^2 by A256,SQUARE_1:15;
                then
A259:           (rm1/2)^2+(rm/2)^2<=(r/2)^2+(r/2)^2 by A258,XREAL_1:7;
                sqrt 2/2<1 by Lm1,SQUARE_1:21,XREAL_1:189;
                then
A260:           r*(sqrt 2/2)<r*1 by A8,XREAL_1:68;
                (r/2)^2 + (r/2)^2= 2*((r/2)^2)
                  .=(sqrt 2)^2*((r/2)^2) by SQUARE_1:def 2
                  .=(r/2*sqrt 2)^2;
                then
A261:           sqrt ((r/2)^2 + (r/2)^2) =r*((sqrt 2)/2) by A8,Lm1,SQUARE_1:22;
A262:           p`1=r2 & p`2=s2 by A207,EUCLID:52;
                0<=(rm/2)^2 & 0<=(rm1/2)^2 by XREAL_1:63;
                then sqrt((rm1/2)^2+(rm/2)^2)<=sqrt((r/2)^2+(r/ 2)^2 ) by A259,
SQUARE_1:26;
                then dist(u,u0) = (Pitag_dist 2).(v,v0) & sqrt ((r2 - r3)^2 +
                (s2 - s3)^2)<r by A260,A261,METRIC_1:def 1,XXREAL_0:2;
                then dist(u,u0) < r by A262,A253,TOPREAL3:7;
                then
A263:           u0 in Ball(u,r) by METRIC_1:11;
                G*(1,j+1)`2-(G*(1,j+1)`2-G*(1,j)`2)/2 >G*(1,j+1)`2-(G*(1
,j+1)`2-G*(1,j)`2)/2 -(G*(1,j+1)`2-G*(1,j)`2)/2 by A255,XREAL_1:44,139;
                then s3>G*(1,j)`2 by A245,A252,XXREAL_0:2;
                then u0 in { |[r1,s1]| : G*(i,1)`1 < r1 & r1 < G*(i+1,1)`1 &
                G*(1,j)`2 < s1 & s1 < G*(1,j+1)`2 } by A257,A249,A251;
                then u0 in Y by A206,GOBOARD6:26;
                hence Y /\ G0 <> {}(TOP-REAL 2) by A9,A263,XBOOLE_0:def 4;
              end;
              case
A264:           G*(i,1)`1 <r2 & r2<G*(i+1,1)`1 & s2=G*(1,j)`2;
                set r3=r2;
                set rl=G*(1,j+1)`2 - G*(1,j)`2;
                set rm=min(r,rl);
                set s3=s2+rm/2;
                reconsider q0=|[r3,s3]| as Point of TOP-REAL 2;
                reconsider u0=q0 as Point of Euclid 2 by EUCLID:22;
A265:           1<=len G by Th34;
                j<j+1 & j+1<=width G by A206,NAT_1:13;
                then G*(1,j)`2<G*(1,j+1)`2 by A206,A265,GOBOARD5:4;
                then
A266:           rl>0 by XREAL_1:50;
                then
A267:           rm>0 by A8,XXREAL_0:21;
                then
A268:           s3>s2 by XREAL_1:29,139;
                rm/2<=rl/2 by XREAL_1:72,XXREAL_0:17;
                then
A269:           G*(1,j)`2+rm/2<=G*(1,j)`2+rl/2 by XREAL_1:6;
                G*(1,j)`2+(G*(1,j+1)`2-G*(1,j)`2)/2 <G*(1,j)`2+(G*(1,j+1
)`2-G*(1,j)`2)/2 +(G*(1,j+1)`2-G*(1,j)`2)/2 by A266,XREAL_1:29,139;
                then s3<G*(1,j+1)`2 by A264,A269,XXREAL_0:2;
                then u0 in { |[r1,s1]| : G*(i,1)`1 < r1 & r1 < G*(i+1,1)`1 &
                G*(1,j)`2 < s1 & s1 < G*(1,j+1)`2 } by A264,A268;
                then
A270:           u0 in Y by A206,GOBOARD6:26;
                sqrt 2/2<1 by Lm1,SQUARE_1:21,XREAL_1:189;
                then
A271:           r*(sqrt 2/2)<r*1 by A8,XREAL_1:68;
                reconsider v0=u0 as Element of REAL 2;
A272:           0<=(rm/2)^2 by XREAL_1:63;
                rm/2<=r/2 by XREAL_1:72,XXREAL_0:17;
                then (rm/2)^2+0^2<=(r/2)^2+0^2 by A267,SQUARE_1:15;
                then
A273:           sqrt(0^2+(rm/2)^2)<=sqrt(0^2+(r/2)^2) by A272,SQUARE_1:26;
A274:           q0`1=r3 & q0`2=s3 by EUCLID:52;
A275:           (r/2)^2 >=0 by XREAL_1:63;
                then 0^2+(r/2)^2 <=(r/2)^2 + (r/2)^2 by XREAL_1:6;
                then
A276:           sqrt
(0^2+(r/2)^2 )<=sqrt((r/2)^2 + (r/2)^2) by A275,SQUARE_1:26;
                (r/2)^2 + (r/2)^2= 2*((r/2)^2)
                  .=(sqrt 2)^2*((r/2)^2) by SQUARE_1:def 2
                  .=(r/2*sqrt 2)^2;
                then sqrt ((r/2)^2 + (r/2)^2) =r*((sqrt 2)/2) by A8,Lm1,
SQUARE_1:22;
                then sqrt(0^2+(r/2)^2)<r by A271,A276,XXREAL_0:2;
                then
A277:           dist(u,u0) = (Pitag_dist 2).(v,v0) & sqrt((r2 - r3)^2 +
                (s2 - s3)^2)<r by A273,METRIC_1:def 1,XXREAL_0:2;
                p`1=r2 & p`2=s2 by A207,EUCLID:52;
                then dist(u,u0) < r by A274,A277,TOPREAL3:7;
                then u0 in Ball(u,r) by METRIC_1:11;
                hence Y /\ G0 <> {}(TOP-REAL 2) by A9,A270,XBOOLE_0:def 4;
              end;
              case
A278:           G*(i,1)`1 <r2 & r2<G*(i+1,1)`1 & G*(1,j)`2 <s2 & s2<
                G*(1,j+1)`2;
                set s3=s2,r3=r2;
                reconsider q0=|[r3,s3]| as Point of TOP-REAL 2;
A279:           q0`1=r3 & q0`2=s3 by EUCLID:52;
                reconsider u0=q0 as Point of Euclid 2 by EUCLID:22;
                reconsider v0=u0 as Element of REAL 2;
                dist(u,u0) = (Pitag_dist 2).(v,v0) & sqrt((r2 - r3)^2 +
                (s2 - s3)^2 )<r by A8,METRIC_1:def 1;
                then dist(u,u0) < r by A207,A279,TOPREAL3:7;
                then
A280:           u0 in Ball(u,r) by METRIC_1:11;
                u0 in { |[r1,s1]| : G*(i,1)`1<r1 & r1<G*(i+1,1)`1 & G*(1
                ,j)`2<s1 & s1<G*(1,j+1)`2} by A278;
                then u0 in Y by A206,GOBOARD6:26;
                hence Y /\ G0 <> {}(TOP-REAL 2) by A9,A280,XBOOLE_0:def 4;
              end;
              case
A281:           G*(i,1)`1 <r2 & r2<G*(i+1,1)`1 & s2=G*(1,j+1)`2;
                set r3=r2;
                set rl=G*(1,j+1)`2 - G*(1,j)`2;
                set rm=min(r,rl);
                set s3=s2-rm/2;
                reconsider q0=|[r3,s3]| as Point of TOP-REAL 2;
                reconsider u0=q0 as Point of Euclid 2 by EUCLID:22;
A282:           1<=len G by Th34;
                j<j+1 & j+1<=width G by A206,NAT_1:13;
                then G*(1,j)`2<G*(1,j+1)`2 by A206,A282,GOBOARD5:4;
                then
A283:           rl>0 by XREAL_1:50;
                then
A284:           rm>0 by A8,XXREAL_0:21;
                then
A285:           s3<s2 by XREAL_1:44,139;
                rm/2<=rl/2 by XREAL_1:72,XXREAL_0:17;
                then
A286:           G*(1,j+1)`2-rm/2>=G*(1,j+1)`2-rl/2 by XREAL_1:13;
                G*(1,j+1)`2-(G*(1,j+1)`2-G*(1,j)`2)/2 >G*(1,j+1)`2-(G*(1
,j+1)`2-G*(1,j)`2)/2 -(G*(1,j+1)`2-G*(1,j)`2)/2 by A283,XREAL_1:44,139;
                then s3>G*(1,j)`2 by A281,A286,XXREAL_0:2;
                then u0 in { |[r1,s1]| : G*(i,1)`1 < r1 & r1 < G*(i+1,1)`1 &
                G*(1,j)`2 < s1 & s1 < G* (1,j+1)`2 } by A281,A285;
                then
A287:           u0 in Y by A206,GOBOARD6:26;
                sqrt 2/2<1 by Lm1,SQUARE_1:21,XREAL_1:189;
                then
A288:           r*(sqrt 2/2)<r*1 by A8,XREAL_1:68;
                reconsider v0=u0 as Element of REAL 2;
A289:           0<=(rm/2)^2 by XREAL_1:63;
                rm/2<=r/2 by XREAL_1:72,XXREAL_0:17;
                then (rm/2)^2+0^2<=(r/2)^2+0^2 by A284,SQUARE_1:15;
                then
A290:           sqrt(0^2+(rm/2)^2)<=sqrt(0^2+(r/2)^2) by A289,SQUARE_1:26;
A291:           q0`1=r3 & q0`2=s3 by EUCLID:52;
A292:           (r/2)^2 >=0 by XREAL_1:63;
                then 0+(r/2)^2 <=(r/2)^2 + (r/2)^2 by XREAL_1:6;
                then
A293:           sqrt
(0^2+(r/2)^2 )<=sqrt((r/2)^2 + (r/2)^2) by A292,SQUARE_1:26;
                (r/2)^2 + (r/2)^2= 2*((r/2)^2)
                  .=(sqrt 2)^2*((r/2)^2) by SQUARE_1:def 2
                  .=(r/2*sqrt 2)^2;
                then sqrt ((r/2)^2 + (r/2)^2) =r*((sqrt 2)/2) by A8,Lm1,
SQUARE_1:22;
                then sqrt(0^2+(r/2)^2)<r by A288,A293,XXREAL_0:2;
                then
A294:           dist(u,u0) = (Pitag_dist 2).(v,v0) & sqrt((r2 - r3)^2 +
                (s2 - s3)^2 )<r by A290,METRIC_1:def 1,XXREAL_0:2;
                p`1=r2 & p`2=s2 by A207,EUCLID:52;
                then dist(u,u0) < r by A291,A294,TOPREAL3:7;
                then u0 in Ball(u,r) by METRIC_1:11;
                hence Y /\ G0 <> {}(TOP-REAL 2) by A9,A287,XBOOLE_0:def 4;
              end;
              case
A295:           r2=G*(i+1,1)`1 & s2=G*(1,j)`2;
                set rl1=G*(i+1,1)`1 - G*(i,1)`1;
                set rl=G*(1,j+1)`2 - G*(1,j)`2;
                set rm=min(r,rl);
                set rm1=min(r,rl1);
                set r3=r2-rm1/2,s3=s2+rm/2;
A296:           1<=width G by Th34;
                i<i+1 & i+1<=len G by A206,NAT_1:13;
                then G*(i,1)`1<G*(i+1,1)`1 by A206,A296,GOBOARD5:3;
                then
A297:           rl1>0 by XREAL_1:50;
                then
A298:           rm1>0 by A8,XXREAL_0:21;
                then r3<r2 by XREAL_1:44,139;
                then
A299:           r3<G*(i+1,1)`1 by A209,XXREAL_0:2;
                rm1/2<=rl1/2 by XREAL_1:72,XXREAL_0:17;
                then
A300:           G*(i+1,1)`1-rm1/2>=G*(i+1,1)`1-rl1/2 by XREAL_1:13;
                G*(i+1,1)`1-(G*(i+1,1)`1-G*(i,1)`1)/2 >G*(i+1,1)`1-(G*(i
+1,1)`1-G*(i,1)`1)/2 -(G*(i+1,1)`1-G*(i,1)`1)/2 by A297,XREAL_1:44,139;
                then
A301:           r3>G*(i,1)`1 by A295,A300,XXREAL_0:2;
                rm/2<=rl/2 by XREAL_1:72,XXREAL_0:17;
                then
A302:           G*(1,j)`2+rm/2<=G*(1,j)`2+rl/2 by XREAL_1:6;
                reconsider q0=|[r3,s3]| as Point of TOP-REAL 2;
A303:           q0`1=r3 & q0`2=s3 by EUCLID:52;
                reconsider u0=q0 as Point of Euclid 2 by EUCLID:22;
                reconsider v0=u0 as Element of REAL 2;
A304:           1<=len G by Th34;
                j<j+1 & j+1<=width G by A206,NAT_1:13;
                then G*(1,j)`2<G*(1,j+1)`2 by A206,A304,GOBOARD5:4;
                then
A305:           rl>0 by XREAL_1:50;
                then
A306:           rm>0 by A8,XXREAL_0:21;
                then s3>s2 by XREAL_1:29,139;
                then
A307:           s3>G*(1,j)`2 by A210,XXREAL_0:2;
                rm1/2<=r/2 by XREAL_1:72,XXREAL_0:17;
                then
A308:           (rm1/2)^2<=(r/2)^2 by A298,SQUARE_1:15;
                rm/2<=r/2 by XREAL_1:72,XXREAL_0:17;
                then (rm/2)^2<=(r/2)^2 by A306,SQUARE_1:15;
                then
A309:           (rm1/2)^2+(rm/2)^2<=(r/2)^2+(r/2)^2 by A308,XREAL_1:7;
                sqrt 2/2<1 by Lm1,SQUARE_1:21,XREAL_1:189;
                then
A310:           r*(sqrt 2/2)<r*1 by A8,XREAL_1:68;
                (r/2)^2 + (r/2)^2= 2*((r/2)^2)
                  .=(sqrt 2)^2*((r/2)^2) by SQUARE_1:def 2
                  .=(r/2*sqrt 2)^2;
                then
A311:           sqrt ((r/2)^2 + (r/2)^2) =r*((sqrt 2)/2) by A8,Lm1,SQUARE_1:22;
A312:           p`1=r2 & p`2=s2 by A207,EUCLID:52;
                0<=(rm/2)^2 & 0<=(rm1/2)^2 by XREAL_1:63;
                then sqrt((rm1/2)^2+(rm/2)^2)<=sqrt((r/2)^2+(r/ 2)^2 ) by A309,
SQUARE_1:26;
                then dist(u,u0) = (Pitag_dist 2).(v,v0) & sqrt ((r2 - r3)^2 +
                (s2 - s3)^2)<r by A310,A311,METRIC_1:def 1,XXREAL_0:2;
                then dist(u,u0) < r by A312,A303,TOPREAL3:7;
                then
A313:           u0 in Ball(u,r) by METRIC_1:11;
                G*(1,j)`2+(G*(1,j+1)`2-G*(1,j)`2)/2 <G*(1,j)`2+(G*(1,j+1
)`2-G*(1,j)`2)/2 +(G*(1,j+1)`2-G*(1,j)`2)/2 by A305,XREAL_1:29,139;
                then s3<G*(1,j+1)`2 by A295,A302,XXREAL_0:2;
                then u0 in { |[r1,s1]| : G*(i,1)`1 < r1 & r1 < G*(i+1,1)`1 &
                G*(1,j)`2 < s1 & s1 < G*(1,j+1)`2 } by A307,A299,A301;
                then u0 in Y by A206,GOBOARD6:26;
                hence Y /\ G0 <> {}(TOP-REAL 2) by A9,A313,XBOOLE_0:def 4;
              end;
              case
A314:           r2=G*(i+1,1)`1 & G*(1,j)`2 <s2 & s2<G*(1,j+1)`2;
                set s3=s2;
                set rl1=G*(i+1,1)`1 - G*(i,1)`1;
                set rm1=min(r,rl1);
                set r3=r2-rm1/2;
                reconsider q0=|[r3,s3]| as Point of TOP-REAL 2;
                reconsider u0=q0 as Point of Euclid 2 by EUCLID:22;
A315:           1<=width G by Th34;
                i<i+1 & i+1<=len G by A206,NAT_1:13;
                then G*(i,1)`1<G*(i+1,1)`1 by A206,A315,GOBOARD5:3;
                then
A316:           rl1>0 by XREAL_1:50;
                then
A317:           rm1>0 by A8,XXREAL_0:21;
                then
A318:           r3<r2 by XREAL_1:44,139;
                rm1/2<=rl1/2 by XREAL_1:72,XXREAL_0:17;
                then
A319:           G*(i+1,1)`1-rm1/2>=G*(i+1,1)`1-rl1/2 by XREAL_1:13;
                G*(i+1,1)`1-(G*(i+1,1)`1-G*(i,1)`1)/2 >G*(i+1,1)`1-(G*(i
+1,1)`1-G*(i,1)`1)/2 -(G*(i+1,1)`1-G*(i,1)`1)/2 by A316,XREAL_1:44,139;
                then r3>G*(i,1)`1 by A314,A319,XXREAL_0:2;
                then u0 in { |[r1,s1]| : G*(i,1)`1 < r1 & r1 < G*(i+1,1)`1 &
                G*(1,j)`2 < s1 & s1 < G* (1,j+1)`2 } by A314,A318;
                then
A320:           u0 in Y by A206,GOBOARD6:26;
                sqrt 2/2<1 by Lm1,SQUARE_1:21,XREAL_1:189;
                then
A321:           r*(sqrt 2/2)<r*1 by A8,XREAL_1:68;
                reconsider v0=u0 as Element of REAL 2;
A322:           0<=(rm1/2)^2 by XREAL_1:63;
                rm1/2<=r/2 by XREAL_1:72,XXREAL_0:17;
                then 0^2+(rm1/2)^2<=0^2+(r/2)^2 by A317,SQUARE_1:15;
                then
A323:           sqrt((rm1/2)^2+0^2)<=sqrt((r/2)^2+0^2) by A322,SQUARE_1:26;
A324:           q0`1=r3 & q0`2=s3 by EUCLID:52;
A325:           (r/2)^2 >=0 by XREAL_1:63;
                then (r/2)^2+0 <=(r/2)^2 + (r/2)^2 by XREAL_1:6;
                then
A326:           sqrt
((r/2)^2+0^2 )<=sqrt((r/2)^2 + (r/2)^2) by A325,SQUARE_1:26;
                (r/2)^2 + (r/2)^2= 2*((r/2)^2)
                  .=(sqrt 2)^2*((r/2)^2) by SQUARE_1:def 2
                  .=(r/2*sqrt 2)^2;
                then sqrt ((r/2)^2 + (r/2)^2) =r*((sqrt 2)/2) by A8,Lm1,
SQUARE_1:22;
                then sqrt((r/2)^2+0^2)<r by A321,A326,XXREAL_0:2;
                then
A327:           dist(u,u0) = (Pitag_dist 2).(v,v0) & sqrt((r2 - r3)^2 +
                (s2 - s3)^2 )<r by A323,METRIC_1:def 1,XXREAL_0:2;
                p`1=r2 & p`2=s2 by A207,EUCLID:52;
                then dist(u,u0) < r by A324,A327,TOPREAL3:7;
                then u0 in Ball(u,r) by METRIC_1:11;
                hence Y /\ G0 <> {}(TOP-REAL 2) by A9,A320,XBOOLE_0:def 4;
              end;
              case
A328:           r2=G*(i+1,1)`1 & s2=G*(1,j+1)`2;
                set rl1=G*(i+1,1)`1 - G*(i,1)`1;
                set rl=G*(1,j+1)`2 - G*(1,j)`2;
                set rm=min(r,rl);
                set rm1=min(r,rl1);
                set r3=r2-rm1/2,s3=s2-rm/2;
A329:           1<=width G by Th34;
                i<i+1 & i+1<=len G by A206,NAT_1:13;
                then G*(i,1)`1<G*(i+1,1)`1 by A206,A329,GOBOARD5:3;
                then
A330:           rl1>0 by XREAL_1:50;
                then
A331:           rm1>0 by A8,XXREAL_0:21;
                then r3<r2 by XREAL_1:44,139;
                then
A332:           r3<G*(i+1,1)`1 by A209,XXREAL_0:2;
                rm1/2<=rl1/2 by XREAL_1:72,XXREAL_0:17;
                then
A333:           G*(i+1,1)`1-rm1/2>=G*(i+1,1)`1-rl1/2 by XREAL_1:13;
                G*(i+1,1)`1-(G*(i+1,1)`1-G*(i,1)`1)/2 >G*(i+1,1)`1-(G*(i
+1,1)`1-G*(i,1)`1)/2 -(G*(i+1,1)`1-G*(i,1)`1)/2 by A330,XREAL_1:44,139;
                then
A334:           r3>G*(i,1)`1 by A328,A333,XXREAL_0:2;
                rm/2<=rl/2 by XREAL_1:72,XXREAL_0:17;
                then
A335:           G*(1,j+1)`2-rm/2>=G*(1,j+1)`2-rl/2 by XREAL_1:13;
                sqrt 2/2<1 by Lm1,SQUARE_1:21,XREAL_1:189;
                then
A336:           r*(sqrt 2/2)<r*1 by A8,XREAL_1:68;
A337:           p`1=r2 & p`2=s2 by A207,EUCLID:52;
                (r/2)^2 + (r/2)^2= 2*((r/2)^2)
                  .=(sqrt 2)^2*((r/2)^2) by SQUARE_1:def 2
                  .=(r/2*sqrt 2)^2;
                then
A338:           sqrt ((r/2)^2 + (r/2)^2) =r*((sqrt 2)/2) by A8,Lm1,SQUARE_1:22;
A339:           1<=len G by Th34;
                rm1/2<=r/2 by XREAL_1:72,XXREAL_0:17;
                then
A340:           (rm1/2)^2<=(r/2)^2 by A331,SQUARE_1:15;
                reconsider q0=|[r3,s3]| as Point of TOP-REAL 2;
A341:           q0`1=r3 & q0`2=s3 by EUCLID:52;
                reconsider u0=q0 as Point of Euclid 2 by EUCLID:22;
                reconsider v0=u0 as Element of REAL 2;
A342:           r2-r3=rm1/2 & s2-s3=rm/2;
                j<j+1 & j+1<=width G by A206,NAT_1:13;
                then G*(1,j)`2<G*(1,j+1)`2 by A206,A339,GOBOARD5:4;
                then
A343:           rl>0 by XREAL_1:50;
                then
A344:           rm>0 by A8,XXREAL_0:21;
                then s3<s2 by XREAL_1:44,139;
                then
A345:           s3<G*(1,j+1)`2 by A211,XXREAL_0:2;
                rm/2<=r/2 by XREAL_1:72,XXREAL_0:17;
                then (rm/2)^2<=(r/2)^2 by A344,SQUARE_1:15;
                then
A346:           (rm1/2)^2+(rm/2)^2<=(r/2)^2+(r/2)^2 by A340,XREAL_1:7;
                0<=(rm/2)^2 & 0<=(rm1/2)^2 by XREAL_1:63;
                then sqrt((rm1/2)^2+(rm/2)^2)<=sqrt((r/2)^2+(r/ 2)^2 ) by A346,
SQUARE_1:26;
                then dist(u,u0) = (Pitag_dist 2).(v,v0) & sqrt ((rm1/2)^2 + (
                rm/2)^2)<r by A336,A338,METRIC_1:def 1,XXREAL_0:2;
                then dist(u,u0) < r by A337,A341,A342,TOPREAL3:7;
                then
A347:           u0 in Ball(u,r) by METRIC_1:11;
                G*(1,j+1)`2-(G*(1,j+1)`2-G*(1,j)`2)/2 >G*(1,j+1)`2-(G*(1
,j+1)`2-G*(1,j)`2)/2 -(G*(1,j+1)`2-G*(1,j)`2)/2 by A343,XREAL_1:44,139;
                then s3>G*(1,j)`2 by A328,A335,XXREAL_0:2;
                then u0 in { |[r1,s1]| : G*(i,1)`1 < r1 & r1 < G*(i+1,1)`1 &
                G*(1,j)`2 < s1 & s1 < G*(1,j+1)`2 } by A345,A332,A334;
                then u0 in Y by A206,GOBOARD6:26;
                hence Y /\ G0 <> {}(TOP-REAL 2) by A9,A347,XBOOLE_0:def 4;
              end;
            end;
            hence Y /\ G0 <> {}(TOP-REAL 2);
          end;
        end;
        hence Y /\ G0 <> {}(TOP-REAL 2);
      end;
      hence thesis;
    end;
    hence thesis by PRE_TOPC:def 7;
  end;
  Cl Y c= Cl cell(G,i,j) & cell(G,i,j) is closed by Th33,PRE_TOPC:19,TOPS_1:16;
  then Cl Y c= cell(G,i,j) by PRE_TOPC:22;
  hence thesis by A2;
end;
