reserve T,T1,T2,S for non empty TopSpace;
reserve p,q for Point of TOP-REAL 2;

theorem Th14:
  for p being Point of TOP-REAL 2 st p<>0.TOP-REAL 2 holds ((p`1<=
p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2)implies Out_In_Sq.p=|[p`1/p`2/p`2,1/p`2
]|) & (not (p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2) implies Out_In_Sq.p=
  |[1/p`1,p`2/p`1/p`1]|)
proof
  let p be Point of TOP-REAL 2;
  assume
A1: p<>0.TOP-REAL 2;
  hereby
    assume
A2: p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2;
    now
      per cases by A2;
      case
A3:     p`1<=p`2 & -p`2<=p`1;
        now
          assume
A4:       p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1;
A5:       now
            per cases by A4;
            case
              p`2<=p`1 & -p`1<=p`2;
              hence p`1=p`2 or p`1=-p`2 by A3,XXREAL_0:1;
            end;
            case
              p`2>=p`1 & p`2<=-p`1;
              then -p`2>=--p`1 by XREAL_1:24;
              hence p`1=p`2 or p`1=-p`2 by A3,XXREAL_0:1;
            end;
          end;
          now
            per cases by A5;
            case
A6:           p`1=p`2;
              then p`1<>0 by A1,EUCLID:53,54;
              then p`1/p`2/p`2=1/p`1 by A6,XCMPLX_1:60;
              hence Out_In_Sq.p=|[p`1/p`2/p`2,1/p`2]| by A1,A4,A6,Def1;
            end;
            case
A7:           p`1=-p`2;
              then
A8:           p`2<>0 by A1,EUCLID:53,54;
A9:           p`1/p`2/p`2=(-(p`2/p`2))/p`2 by A7
                .=(-1)/p`2 by A8,XCMPLX_1:60
                .= 1/p`1 by A7,XCMPLX_1:192;
              -p`1=p`2 by A7;
              then 1/p`2= -(1/p`1) by XCMPLX_1:188
                .=-(p`2/p`1/(-p`1)) by A7,A9,XCMPLX_1:192
                .=--(p`2/p`1/p`1) by XCMPLX_1:188
                .=p`2/p`1/p`1;
              hence Out_In_Sq.p=|[p`1/p`2/p`2,1/p`2]| by A1,A4,A9,Def1;
            end;
          end;
          hence Out_In_Sq.p=|[p`1/p`2/p`2,1/p`2]|;
        end;
        hence Out_In_Sq.p=|[p`1/p`2/p`2,1/p`2]| by A1,Def1;
      end;
      case
A10:    p`1>=p`2 & p`1<=-p`2;
        now
          assume
A11:      p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1;
A12:      now
            per cases by A11;
            case
              p`2<=p`1 & -p`1<=p`2;
              then --p`1>=-p`2 by XREAL_1:24;
              hence p`1=p`2 or p`1=-p`2 by A10,XXREAL_0:1;
            end;
            case
              p`2>=p`1 & p`2<=-p`1;
              hence p`1=p`2 or p`1=-p`2 by A10,XXREAL_0:1;
            end;
          end;
          now
            per cases by A12;
            case
A13:          p`1=p`2;
              then p`1 <> 0 by A1,EUCLID:53,54;
              then p`1/p`2/p`2=1/p`1 by A13,XCMPLX_1:60;
              hence Out_In_Sq.p=|[p`1/p`2/p`2,1/p`2]| by A1,A11,A13,Def1;
            end;
            case
A14:          p`1=-p`2;
              then
A15:          p`2<>0 by A1,EUCLID:53,54;
A16:          p`1/p`2/p`2 =(-(p`2/p`2))/p`2 by A14
                .=(-1)/p`2 by A15,XCMPLX_1:60
                .= 1/p`1 by A14,XCMPLX_1:192;
              -p`1=p`2 by A14;
              then 1/p`2=-(p`1/p`2/p`2) by A16,XCMPLX_1:188
                .=-(p`2/p`1/(-p`1)) by A14,XCMPLX_1:191
                .=--(p`2/p`1/p`1) by XCMPLX_1:188
                .=p`2/p`1/p`1;
              hence Out_In_Sq.p=|[p`1/p`2/p`2,1/p`2]| by A1,A11,A16,Def1;
            end;
          end;
          hence Out_In_Sq.p=|[p`1/p`2/p`2,1/p`2]|;
        end;
        hence Out_In_Sq.p=|[p`1/p`2/p`2,1/p`2]| by A1,Def1;
      end;
    end;
    hence Out_In_Sq.p=|[p`1/p`2/p`2,1/p`2]|;
  end;
  hereby
A17: -p`2>p`1 implies --p`2<-p`1 by XREAL_1:24;
A18: -p`2<p`1 implies --p`2>-p`1 by XREAL_1:24;
    assume not(p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2);
    hence Out_In_Sq.p=|[1/p`1,p`2/p`1/p`1]| by A1,A18,A17,Def1;
  end;
end;
