reserve x for Real;
reserve p,q for Point of TOP-REAL 2;

theorem Th22:
  Sq_Circ is one-to-one
proof
  let x1,x2 be object;
  assume that
A1: x1 in dom Sq_Circ and
A2: x2 in dom Sq_Circ and
A3: Sq_Circ.x1=Sq_Circ.x2;
  reconsider p2=x2 as Point of TOP-REAL 2 by A2;
  reconsider p1=x1 as Point of TOP-REAL 2 by A1;
  set q=p1,p=p2;
  per cases;
  suppose
A4: q=0.TOP-REAL 2;
    then
A5: Sq_Circ.q=0.TOP-REAL 2 by Def1;
    now
      per cases;
      case
        p=0.TOP-REAL 2;
        hence thesis by A4;
      end;
      case
A6:     p<>0.TOP-REAL 2 & (p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p `1);
        (p`2/p`1)^2 >=0 by XREAL_1:63;
        then 1+(p`2/p`1)^2>=1+0 by XREAL_1:7;
        then
A7:     sqrt(1+(p`2/p`1)^2)>=1 by SQUARE_1:18,26;
A8:     Sq_Circ.p=|[p`1/sqrt(1+(p`2/p`1)^2),p`2/sqrt(1+(p`2/p`1)^2)]| by A6
,Def1;
        then p`2/sqrt(1+(p`2/p`1)^2)=0 by A3,A5,EUCLID:52,JGRAPH_2:3;
        then
A9:     p`2= 0 *sqrt(1+(p`2/p`1)^2) by A7,XCMPLX_1:87
          .=0;
        p`1/sqrt(1+(p`2/p`1)^2)=0 by A3,A5,A8,EUCLID:52,JGRAPH_2:3;
        then p`1= 0 *sqrt(1+(p`2/p`1)^2) by A7,XCMPLX_1:87
          .=0;
        hence contradiction by A6,A9,EUCLID:53,54;
      end;
      case
A10:    p<>0.TOP-REAL 2 & not (p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2 <=-p`1);
        (p`1/p`2)^2 >=0 by XREAL_1:63;
        then 1+(p`1/p`2)^2>=1+0 by XREAL_1:7;
        then
A11:    sqrt(1+(p`1/p`2)^2)>=1 by SQUARE_1:18,26;
        Sq_Circ.p=|[p`1/sqrt(1+(p`1/p`2)^2),p`2/sqrt(1+(p`1/p`2)^2)]| by A10
,Def1;
        then p`2/sqrt(1+(p`1/p`2)^2)=0 by A3,A5,EUCLID:52,JGRAPH_2:3;
        then p`2= 0 *sqrt(1+(p`1/p`2)^2) by A11,XCMPLX_1:87
          .=0;
        hence contradiction by A10;
      end;
    end;
    hence thesis;
  end;
  suppose
A12: q<>0.TOP-REAL 2 & (q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q`1);
A13: sqrt(1+(q`2/q`1)^2)>0 by Lm1,SQUARE_1:25;
A14: Sq_Circ.q=|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]| by A12,Def1;
A15: (|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|)`2 = q`2/sqrt(1+
    (q `2/q`1)^2) by EUCLID:52;
A16: 1+(q`2/q`1)^2>0 by Lm1;
A17: (|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|)`1 = q`1/sqrt(1+
    (q `2/q`1)^2) by EUCLID:52;
    now
      per cases;
      case
A18:    p=0.TOP-REAL 2;
        (q`2/q`1)^2 >=0 by XREAL_1:63;
        then 1+(q`2/q`1)^2>=1+0 by XREAL_1:7;
        then
A19:    sqrt(1+(q`2/q`1)^2)>=1 by SQUARE_1:18,26;
A20:    Sq_Circ.p=0.TOP-REAL 2 by A18,Def1;
        then q`2/sqrt(1+(q`2/q`1)^2)=0 by A3,A14,EUCLID:52,JGRAPH_2:3;
        then
A21:    q`2= 0 *sqrt(1+(q`2/q`1)^2) by A19,XCMPLX_1:87
          .=0;
        q`1/sqrt(1+(q`2/q`1)^2)=0 by A3,A14,A20,EUCLID:52,JGRAPH_2:3;
        then q`1= 0 *sqrt(1+(q`2/q`1)^2) by A19,XCMPLX_1:87
          .=0;
        hence contradiction by A12,A21,EUCLID:53,54;
      end;
      case
A22:    p<>0.TOP-REAL 2 & (p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p `1);
        now
          assume
A23:      p`1=0;
          then p`2=0 by A22;
          hence contradiction by A22,A23,EUCLID:53,54;
        end;
        then
A24:    (p`1)^2>0 by SQUARE_1:12;
A25:    sqrt(1+(p`2/p`1)^2)>0 by Lm1,SQUARE_1:25;
A26:    1+(p`2/p`1)^2>0 by Lm1;
A27:    Sq_Circ.p=|[p`1/sqrt(1+(p`2/p`1)^2),p`2/sqrt(1+(p`2/p`1)^2 )]| by A22
,Def1;
        then
A28:    p
`2/sqrt(1+(p`2/p`1)^2)=q`2/sqrt(1+(q`2/q`1)^2) by A3,A14,A15,EUCLID:52;
        then (p`2)^2/(sqrt(1+(p`2/p`1)^2))^2=(q`2/sqrt(1+(q`2/q`1)^2))^2 by
XCMPLX_1:76;
        then (p`2)^2/(sqrt(1+(p`2/p`1)^2))^2=(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2
        by XCMPLX_1:76;
        then (p`2)^2/(1+(p`2/p`1)^2)=(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2 by A26,
SQUARE_1:def 2;
        then
A29:    (p`2)^2/(1+(p`2/p`1)^2)=(q`2)^2/(1+(q`2/q`1)^2) by A16,SQUARE_1:def 2;
A30:    p`1/sqrt(1+(p`2/p`1)^2)=q`1/sqrt(1+(q`2/q`1)^2) by A3,A14,A17,A27,
EUCLID:52;
        then (p`1)^2/(sqrt(1+(p`2/p`1)^2))^2=(q`1/sqrt(1+(q`2/q`1)^2))^2 by
XCMPLX_1:76;
        then (p`1)^2/(sqrt(1+(p`2/p`1)^2))^2=(q`1)^2/(sqrt(1+(q`2/q`1)^2))^2
        by XCMPLX_1:76;
        then (p`1)^2/(1+(p`2/p`1)^2)=(q`1)^2/(sqrt(1+(q`2/q`1)^2))^2 by A26,
SQUARE_1:def 2;
        then (p`1)^2/(1+(p`2/p`1)^2)=(q`1)^2/(1+(q`2/q`1)^2) by A16,
SQUARE_1:def 2;
        then (p`1)^2/(1+(p`2/p`1)^2)/(p`1)^2=(q`1)^2/(p`1)^2/(1+(q`2/q`1)^2)
        by XCMPLX_1:48;
        then (p`1)^2/(p`1)^2/(1+(p`2/p`1)^2)=(q`1)^2/(p`1)^2/(1+(q`2/q`1)^2)
        by XCMPLX_1:48;
        then 1/(1+(p`2/p`1)^2)=(q`1)^2/(p`1)^2/(1+(q`2/q`1)^2) by A24,
XCMPLX_1:60;
        then
A31:    1/(1+(p`2/p`1)^2)*(1+(q`2/q`1)^2)=(q`1)^2/(p`1)^2 by A16,XCMPLX_1:87;
        now
          assume
A32:      q`1=0;
          then q`2=0 by A12;
          hence contradiction by A12,A32,EUCLID:53,54;
        end;
        then
A33:    (q`1)^2>0 by SQUARE_1:12;
        now
          per cases;
          case
A34:        p`2=0;
            then (q`2)^2=0 by A16,A29,XCMPLX_1:50;
            then
A35:        q`2=0 by XCMPLX_1:6;
            then p=|[q`1,0]|by A3,A14,A27,A34,EUCLID:53;
            hence thesis by A35,EUCLID:53;
          end;
          case
            p`2<>0;
            then
A36:        (p`2)^2>0 by SQUARE_1:12;
            (p`2)^2/(1+(p`2/p`1)^2)/(p`2)^2=(q`2)^2/(p`2)^2/(1+(q`2/q`1)
            ^2) by A29,XCMPLX_1:48;
            then (p`2)^2/(p`2)^2/(1+(p`2/p`1)^2)=(q`2)^2/(p`2)^2/(1+(q`2/q`1)
            ^2) by XCMPLX_1:48;
            then 1/(1+(p`2/p`1)^2)=(q`2)^2/(p`2)^2/(1+(q`2/q`1)^2) by A36,
XCMPLX_1:60;
            then 1/(1+(p`2/p`1)^2)*(1+(q`2/q`1)^2)=(q`2)^2/(p`2)^2 by A16,
XCMPLX_1:87;
            then (q`1)^2/(q`1)^2/(p`1)^2=(q`2)^2/(p`2)^2/(q`1)^2 by A31,
XCMPLX_1:48;
            then 1/(p`1)^2=(q`2)^2/(p`2)^2/(q`1)^2 by A33,XCMPLX_1:60;
            then 1/(p`1)^2*(p`2)^2=(p`2)^2*((q`2)^2/(p`2)^2)/(q`1)^2 by
XCMPLX_1:74;
            then 1/(p`1)^2*(p`2)^2=(q`2)^2/(q`1)^2 by A36,XCMPLX_1:87;
            then (p`2)^2/(p`1)^2=(q`2)^2/(q`1)^2 by XCMPLX_1:99;
            then (p`2/p`1)^2=(q`2)^2/(q`1)^2 by XCMPLX_1:76;
            then
A37:        (1+(p`2/p`1)^2)=(1+(q`2/q`1)^2) by XCMPLX_1:76;
            then p`2=q`2/sqrt(1+(q`2/q`1)^2)*sqrt(1+(q`2/q`1)^2) by A28,A25,
XCMPLX_1:87;
            then
A38:        p`2=q`2 by A13,XCMPLX_1:87;
            p`1=q`1/sqrt(1+(q`2/q`1)^2)*sqrt(1+(q`2/q`1)^2) by A30,A25,A37,
XCMPLX_1:87;
            then p`1=q`1 by A13,XCMPLX_1:87;
            then p=|[q`1,q`2]|by A38,EUCLID:53;
            hence thesis by EUCLID:53;
          end;
        end;
        hence thesis;
      end;
      case
A39:    p<>0.TOP-REAL 2 & not (p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2 <=-p`1);
A40:    1+(p`1/p`2)^2>0 by Lm1;
A41:    p<>0.TOP-REAL 2 & p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2 by A39,
JGRAPH_2:13;
        p`2<>0 by A39;
        then
A42:    (p`2)^2>0 by SQUARE_1:12;
        (|[p`1/sqrt(1+(p`1/p`2)^2),p`2/sqrt(1+(p`1/p`2)^2)]|)`2 = p`2/
        sqrt(1+(p `1/p`2)^2) by EUCLID:52;
        then
A43:    p`2/sqrt(1+(p`1/p`2)^2)=q`2/sqrt(1+(q`2/q`1)^2) by A3,A14,A15,A39,Def1;
        then (p`2)^2/(sqrt(1+(p`1/p`2)^2))^2=(q`2/sqrt(1+(q`2/q`1)^2))^2 by
XCMPLX_1:76;
        then (p`2)^2/(sqrt(1+(p`1/p`2)^2))^2=(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2
        by XCMPLX_1:76;
        then (p`2)^2/(1+(p`1/p`2)^2)=(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2 by A40,
SQUARE_1:def 2;
        then (p`2)^2/(1+(p`1/p`2)^2)=(q`2)^2/(1+(q`2/q`1)^2) by A16,
SQUARE_1:def 2;
        then (p`2)^2/(1+(p`1/p`2)^2)/(p`2)^2=(q`2)^2/(p`2)^2/(1+(q`2/q`1)^2)
        by XCMPLX_1:48;
        then (p`2)^2/(p`2)^2/(1+(p`1/p`2)^2)=(q`2)^2/(p`2)^2/(1+(q`2/q`1)^2)
        by XCMPLX_1:48;
        then 1/(1+(p`1/p`2)^2)=(q`2)^2/(p`2)^2/(1+(q`2/q`1)^2) by A42,
XCMPLX_1:60;
        then
A44:    1/(1+(p`1/p`2)^2)*(1+(q`2/q`1)^2)=(q`2)^2/(p`2)^2 by A16,XCMPLX_1:87;
A45:    sqrt(1+(p`1/p`2)^2)>0 by Lm1,SQUARE_1:25;
        (|[p`1/sqrt(1+(p`1/p`2)^2),p`2/sqrt(1+(p`1/p`2)^2)]|)`1 = p`1/
        sqrt(1+(p `1/p`2)^2) by EUCLID:52;
        then
A46:    p`1/sqrt(1+(p`1/p`2)^2)=q`1/sqrt(1+(q`2/q`1)^2) by A3,A14,A17,A39,Def1;
        then (p`1)^2/(sqrt(1+(p`1/p`2)^2))^2=(q`1/sqrt(1+(q`2/q`1)^2))^2 by
XCMPLX_1:76;
        then (p`1)^2/(sqrt(1+(p`1/p`2)^2))^2=(q`1)^2/(sqrt(1+(q`2/q`1)^2))^2
        by XCMPLX_1:76;
        then (p`1)^2/(1+(p`1/p`2)^2)=(q`1)^2/(sqrt(1+(q`2/q`1)^2))^2 by A40,
SQUARE_1:def 2;
        then
A47:    (p`1)^2/(1+(p`1/p`2)^2)=(q`1)^2/(1+(q`2/q`1)^2) by A16,SQUARE_1:def 2;
A48:    now
          assume
A49:      q`1=0;
          then q`2=0 by A12;
          hence contradiction by A12,A49,EUCLID:53,54;
        end;
        then
A50:    (q`1)^2>0 by SQUARE_1:12;
        now
          per cases;
          case
            p`1=0;
            then (q`1)^2=0 by A16,A47,XCMPLX_1:50;
            then
A51:        q`1=0 by XCMPLX_1:6;
            then q`2=0 by A12;
            hence contradiction by A12,A51,EUCLID:53,54;
          end;
          case
A52:        p`1<>0;
            set a=q`2/q`1;
            (p`1)^2/(1+(p`1/p`2)^2)/(p`1)^2=(q`1)^2/(p`1)^2/(1+(q`2/q`1)
            ^2) by A47,XCMPLX_1:48;
            then
A53:        (p`1)^2/(p`1)^2/(1+(p`1/p`2)^2)=(q`1)^2/(p`1)^2/(1+(q`2/q`1)
            ^2) by XCMPLX_1:48;
A54:        q`1*a<=q`1 & -q`1<=q`1*a or q`1*a>=q`1 & q`1*a<=-q`1 by A12,A48,
XCMPLX_1:87;
A55:        now
              per cases by A48;
              case
A56:            q`1>0;
                then a*q`1/q`1<=q`1/q`1 & (-q`1)/q`1<=a*q`1/q`1 or a*q`1/q`1
                >=q`1/q`1 & a*q`1/q`1<=(-q`1)/q`1 by A54,XREAL_1:72;
                then
A57:            a<=q`1/q`1 & (-q`1)/q`1<=a or a>=q`1/q`1 & a<=(-q`1)/q`1
                by A56,XCMPLX_1:89;
                q`1/q`1=1 by A56,XCMPLX_1:60;
                hence a<=1 & -1<=a or a>=1 & a<=-1 by A57,XCMPLX_1:187;
              end;
              case
A58:            q`1<0;
                then
A59:            q`1/q`1=1 & (-q`1)/q`1=-1 by XCMPLX_1:60,197;
                a*q`1/q`1>=q`1/q`1 & (-q`1)/q`1>=a*q`1/q`1 or a*q`1/q`1
                <=q`1/q`1 & a*q`1/q`1>=(-q`1)/q`1 by A54,A58,XREAL_1:73;
                hence a<=1 & -1<=a or a>=1 & a<=-1 by A58,A59,XCMPLX_1:89;
              end;
            end;
            (p`1)^2>0 by A52,SQUARE_1:12;
            then 1/(1+(p`1/p`2)^2)=(q`1)^2/(p`1)^2/(1+(q`2/q`1)^2) by A53,
XCMPLX_1:60;
            then 1/(1+(p`1/p`2)^2)*(1+(q`2/q`1)^2)=(q`1)^2/(p`1)^2 by A16,
XCMPLX_1:87;
            then
(q`1)^2/(q`1)^2/(p`1)^2=(q`2)^2/(p`2)^2/(q`1)^2 by A44,XCMPLX_1:48;
            then 1/(p`1)^2=(q`2)^2/(p`2)^2/(q`1)^2 by A50,XCMPLX_1:60;
            then 1/(p`1)^2*(p`2)^2=(p`2)^2*((q`2)^2/(p`2)^2)/(q`1)^2 by
XCMPLX_1:74;
            then 1/(p`1)^2*(p`2)^2=(q`2)^2/(q`1)^2 by A42,XCMPLX_1:87;
            then (p`2)^2/(p`1)^2=(q`2)^2/(q`1)^2 by XCMPLX_1:99;
            then (p`2/p`1)^2=(q`2)^2/(q`1)^2 by XCMPLX_1:76;
            then
A60:        (p`2/p`1)^2=(q`2/q`1)^2 by XCMPLX_1:76;
            then
A61:        p`2/p`1*p`1=a*p`1 or p`2/p`1*p`1=(-a)*p`1 by SQUARE_1:40;
A62:        now
              per cases by A52,A61,XCMPLX_1:87;
              case
A63:            p`2=a*p`1;
                now
                  per cases by A52;
                  case
                    p`1>0;
                    then p`1/p`1<= a*p`1/p`1 & (-(a*p`1))/p`1<=p`1/p`1 or p`1
/p`1>=(a*p`1)/p`1 & p`1/p`1<=(-(a*p`1))/p`1 by A41,A63,XREAL_1:72;
                    then
A64:                1<= a*p`1/p`1 & (-(a*p`1))/p`1<=1 or 1>=(a*p`1)/p`1
                    & 1<=(-(a*p`1))/p`1 by A52,XCMPLX_1:60;
                    (a*p`1)/p`1=a by A52,XCMPLX_1:89;
                    hence 1<=a & -a<=1 or 1>=a & 1<=-a by A64,XCMPLX_1:187;
                  end;
                  case
                    p`1<0;
                    then p`1/p`1>= a*p`1/p`1 & (-(a*p`1))/p`1>=p`1/p`1 or p`1
/p`1<=(a*p`1)/p`1 & p`1/p`1>=(-(a*p`1))/p`1 by A41,A63,XREAL_1:73;
                    then
A65:                1>= a*p`1/p`1 & (-(a*p`1))/p`1>=1 or 1<=(a*p`1)/p`1
                    & 1>=(-(a*p`1))/p`1 by A52,XCMPLX_1:60;
                    (a*p`1)/p`1=a by A52,XCMPLX_1:89;
                    hence 1<=a & -a<=1 or 1>=a & 1<=-a by A65,XCMPLX_1:187;
                  end;
                end;
                then 1<=a & -a<=1 or 1>=a & -1>=--a by XREAL_1:24;
                hence 1<=a or -1>=a;
              end;
              case
A66:            p`2=(-a)*p`1;
                now
                  per cases by A52;
                  case
                    p`1>0;
                    then p`1/p`1<= (-a)*p`1/p`1 & (-((-a)*p`1))/p`1<=p`1/p`1
or p`1/p`1>=((-a)*p`1)/p`1 & p`1/p`1<=(-((-a)*p`1))/p`1 by A41,A66,XREAL_1:72;
                    then 1<= (-a)*p`1/p`1 & (-((-a)*p`1))/p `1<=1 or 1>=((-a)
                    *p`1)/p`1 & 1<=(-((-a)*p`1))/p`1 by A52,XCMPLX_1:60;
                    then
A67:                1<= (-a) & -(((-a)*p`1)/p`1)<=1 or 1>=(-a) & 1<=-(((
                    -a)*p`1)/p`1) by A52,XCMPLX_1:89,187;
                    ((-a)*p`1)/p`1=(-a) by A52,XCMPLX_1:89;
                    hence 1<=a & -a<=1 or 1>=a & 1<=-a by A67;
                  end;
                  case
                    p`1<0;
                    then p`1/p`1>= (-a)*p`1/p`1 & (-((-a)*p`1))/p`1>=p`1/p`1
or p`1/p`1<=((-a)*p`1)/p`1 & p`1/p`1>=(-((-a)*p`1))/p`1 by A41,A66,XREAL_1:73;
                    then 1>= (-a)*p`1/p`1 & (-((-a)*p`1))/p `1>=1 or 1<=((-a)
                    *p`1)/p`1 & 1>=(-((-a)*p`1))/p`1 by A52,XCMPLX_1:60;
                    then
A68:                1>= (-a) & -(((-a)*p`1)/p`1)>=1 or 1<=(-a) & 1>=-(((
                    -a)*p`1)/p`1) by A52,XCMPLX_1:89,187;
                    ((-a)*p`1)/p`1=(-a) by A52,XCMPLX_1:89;
                    hence 1<=a & -a<=1 or 1>=a & 1<=-a by A68;
                  end;
                end;
                then 1<=a & -a<=1 or 1>=a & -1>=--a by XREAL_1:24;
                hence 1<=a or -1>=a;
              end;
            end;
A69:        now
              per cases by A62,A55,XXREAL_0:1;
              case
                a=1;
                then (p`2)^2/(p`1)^2=1 by A60,XCMPLX_1:76;
                then
A70:            (p`2)^2=(p`1)^2 by XCMPLX_1:58;
                (p`1/p`2)^2=(p`1)^2/(p`2)^2 by XCMPLX_1:76;
                hence (p`1/p`2)^2=(q`2/q`1)^2 by A60,A70,XCMPLX_1:76;
              end;
              case
                a=-1;
                then (p`2)^2/(p`1)^2=1 by A60,XCMPLX_1:76;
                then
A71:            (p`2)^2=(p`1)^2 by XCMPLX_1:58;
                (p`1/p`2)^2=(p`1)^2/(p`2)^2 by XCMPLX_1:76;
                hence (p`1/p`2)^2=(q`2/q`1)^2 by A60,A71,XCMPLX_1:76;
              end;
            end;
            then p`2=q`2/sqrt(1+(q`2/q`1)^2)*sqrt(1+(q`2/q`1)^2) by A43,A45,
XCMPLX_1:87;
            then
A72:        p`2=q`2 by A13,XCMPLX_1:87;
            p`1=q`1/sqrt(1+(q`2/q`1)^2)*sqrt(1+(q`2/q`1)^2) by A46,A45,A69,
XCMPLX_1:87;
            then p`1=q`1 by A13,XCMPLX_1:87;
            then p=|[q`1,q`2]| by A72,EUCLID:53;
            hence thesis by EUCLID:53;
          end;
        end;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
  suppose
A73: q<>0.TOP-REAL 2 & not (q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=- q`1);
A74: (|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2)]|)`2 = q`2/sqrt(1
    +( q`1/q`2)^2) by EUCLID:52;
A75: (|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2)]|)`1 = q`1/sqrt(1
    +( q`1/q`2)^2) by EUCLID:52;
A76: 1+(q`1/q`2)^2>0 by Lm1;
A77: sqrt(1+(q`1/q`2)^2)>0 by Lm1,SQUARE_1:25;
A78: Sq_Circ.q=|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2)]| by A73,Def1;
A79: q`1<=q`2 & -q`2<=q`1 or q`1>=q`2 & q`1<=-q`2 by A73,JGRAPH_2:13;
    now
      per cases;
      case
A80:    p=0.TOP-REAL 2;
        (q`1/q`2)^2 >=0 by XREAL_1:63;
        then 1+(q`1/q`2)^2>=1+0 by XREAL_1:7;
        then
A81:    sqrt(1+(q`1/q`2)^2)>=1 by SQUARE_1:18,26;
        Sq_Circ.p=0.TOP-REAL 2 by A80,Def1;
        then q`2/sqrt(1+(q`1/q`2)^2)=0 by A3,A78,EUCLID:52,JGRAPH_2:3;
        then q`2= 0 *sqrt(1+(q`1/q`2)^2) by A81,XCMPLX_1:87
          .=0;
        hence contradiction by A73;
      end;
      case
A82:    p<>0.TOP-REAL 2 & (p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=- p`1);
        now
          assume
A83:      p`1=0;
          then p`2=0 by A82;
          hence contradiction by A82,A83,EUCLID:53,54;
        end;
        then
A84:    (p`1)^2>0 by SQUARE_1:12;
A85:    1+(p`2/p`1)^2>0 by Lm1;
A86:    Sq_Circ.p=|[p`1/sqrt(1+(p`2/p`1)^2),p`2/sqrt(1+(p`2/p`1)^2)]| by A82
,Def1;
        then
A87:    p`1/sqrt(1+(p`2/p`1)^2)=q`1/sqrt(1+(q`1/q`2)^2) by A3,A78,A75,EUCLID:52
;
        then (p`1)^2/(sqrt(1+(p`2/p`1)^2))^2=(q`1/sqrt(1+(q`1/q`2)^2))^2 by
XCMPLX_1:76;
        then (p`1)^2/(sqrt(1+(p`2/p`1)^2))^2=(q`1)^2/(sqrt(1+(q`1/q`2)^2))^2
        by XCMPLX_1:76;
        then (p`1)^2/(1+(p`2/p`1)^2)=(q`1)^2/(sqrt(1+(q`1/q`2)^2))^2 by A85,
SQUARE_1:def 2;
        then (p`1)^2/(1+(p`2/p`1)^2)=(q`1)^2/(1+(q`1/q`2)^2) by A76,
SQUARE_1:def 2;
        then (p`1)^2/(1+(p`2/p`1)^2)/(p`1)^2=(q`1)^2/(p`1)^2/(1+(q`1/q`2)^2)
        by XCMPLX_1:48;
        then (p`1)^2/(p`1)^2/(1+(p`2/p`1)^2)=(q`1)^2/(p`1)^2/(1+(q`1/q`2)^2)
        by XCMPLX_1:48;
        then 1/(1+(p`2/p`1)^2)=(q`1)^2/(p`1)^2/(1+(q`1/q`2)^2) by A84,
XCMPLX_1:60;
        then
A88:    1/(1+(p`2/p`1)^2)*(1+(q`1/q`2)^2)=(q`1)^2/(p`1)^2 by A76,XCMPLX_1:87;
A89:    p`2/sqrt(1+(p`2/p`1)^2)=q`2/sqrt(1+(q`1/q`2)^2) by A3,A78,A74,A86,
EUCLID:52;
        then (p`2)^2/(sqrt(1+(p`2/p`1)^2))^2=(q`2/sqrt(1+(q`1/q`2)^2))^2 by
XCMPLX_1:76;
        then (p`2)^2/(sqrt(1+(p`2/p`1)^2))^2=(q`2)^2/(sqrt(1+(q`1/q`2)^2))^2
        by XCMPLX_1:76;
        then (p`2)^2/(1+(p`2/p`1)^2)=(q`2)^2/(sqrt(1+(q`1/q`2)^2))^2 by A85,
SQUARE_1:def 2;
        then
A90:    (p`2)^2/(1+(p`2/p`1)^2)=(q`2)^2/(1+(q`1/q`2)^2) by A76,SQUARE_1:def 2;
A91:    sqrt(1+(p`2/p`1)^2)>0 by Lm1,SQUARE_1:25;
A92:    q`2<>0 by A73;
        then
A93:    (q`2)^2>0 by SQUARE_1:12;
        now
          per cases;
          case
            p`2=0;
            then (q`2)^2=0 by A76,A90,XCMPLX_1:50;
            then q`2=0 by XCMPLX_1:6;
            hence contradiction by A73;
          end;
          case
A94:        p`2<>0;
            set a=q`1/q`2;
            (p`2)^2/(1+(p`2/p`1)^2)/(p`2)^2=(q`2)^2/(p`2)^2/(1+(q`1/q`2)
            ^2) by A90,XCMPLX_1:48;
            then
A95:        (p`2)^2/(p`2)^2/(1+(p`2/p`1)^2)=(q`2)^2/(p`2)^2/(1+(q`1/q`2)
            ^2) by XCMPLX_1:48;
A96:        q`2*a<=q`2 & -q`2<=q`2*a or q`2*a>=q`2 & q`2*a<=-q`2 by A79,A92,
XCMPLX_1:87;
A97:        now
              per cases by A73;
              case
A98:            q`2>0;
                then
A99:            q`2/q`2=1 & (-q`2)/q`2=-1 by XCMPLX_1:60,197;
                a*q`2/q`2<=q`2/q`2 & (-q`2)/q`2<=a*q`2/q`2 or a*q`2/q`2
                >=q`2/q`2 & a*q`2/q`2<=(-q`2)/q`2 by A96,A98,XREAL_1:72;
                hence a<=1 & -1<=a or a>=1 & a<=-1 by A98,A99,XCMPLX_1:89;
              end;
              case
A100:           q`2<0;
                then a*q`2/q`2>=q`2/q`2 & (-q`2)/q`2>=a*q`2/q`2 or a*q`2/q`2
                <=q`2/q`2 & a*q`2/q`2>=(-q`2)/q`2 by A96,XREAL_1:73;
                then a>=q`2/q`2 & (-q`2)/q`2>=a or a<=q`2/q`2 & a>=(-q`2)/q`2
                by A100,XCMPLX_1:89;
                hence a<=1 & -1<=a or a>=1 & a<=-1 by A100,XCMPLX_1:60,197;
              end;
            end;
            (p`2)^2>0 by A94,SQUARE_1:12;
            then 1/(1+(p`2/p`1)^2)=(q`2)^2/(p`2)^2/(1+(q`1/q`2)^2) by A95,
XCMPLX_1:60;
            then 1/(1+(p`2/p`1)^2)*(1+(q`1/q`2)^2)=(q`2)^2/(p`2)^2 by A76,
XCMPLX_1:87;
            then
            (q`2)^2/(q`2)^2/(p`2)^2=(q`1)^2/(p`1)^2/(q`2)^2 by A88,XCMPLX_1:48;
            then 1/(p`2)^2=(q`1)^2/(p`1)^2/(q`2)^2 by A93,XCMPLX_1:60;
            then 1/(p`2)^2*(p`1)^2=(p`1)^2*((q`1)^2/(p`1)^2)/(q`2)^2 by
XCMPLX_1:74;
            then 1/(p`2)^2*(p`1)^2=(q`1)^2/(q`2)^2 by A84,XCMPLX_1:87;
            then (p`1)^2/(p`2)^2=(q`1)^2/(q`2)^2 by XCMPLX_1:99;
            then (p`1/p`2)^2=(q`1)^2/(q`2)^2 by XCMPLX_1:76;
            then
A101:       (p`1/p`2)^2=(q`1/q`2)^2 by XCMPLX_1:76;
            then
A102:       p`1/p`2=q`1/q`2 or p`1/p`2=-q`1/q`2 by SQUARE_1:40;
A103:       now
              per cases by A94,A102,XCMPLX_1:87;
              case
A104:           p`1=a*p`2;
                now
                  per cases by A94;
                  case
                    p`2>0;
                    then p`2/p`2<= a*p`2/p`2 & (-(a*p`2))/p`2<=p`2/p`2 or p`2
/p`2>=(a*p`2)/p`2 & p`2/p`2<=(-(a*p`2))/p`2 by A82,A104,XREAL_1:72;
                    then
A105:               1<= a*p`2/p`2 & (-(a*p`2))/p`2<=1 or 1>=(a*p`2)/p`2
                    & 1<=(-(a*p`2))/p`2 by A94,XCMPLX_1:60;
                    (a*p`2)/p`2=a by A94,XCMPLX_1:89;
                    hence 1<=a & -a<=1 or 1>=a & 1<=-a by A105,XCMPLX_1:187;
                  end;
                  case
                    p`2<0;
                    then p`2/p`2>= a*p`2/p`2 & (-(a*p`2))/p`2>=p`2/p`2 or p`2
/p`2<=(a*p`2)/p`2 & p`2/p`2>=(-(a*p`2))/p`2 by A82,A104,XREAL_1:73;
                    then
A106:               1>= a*p`2/p`2 & (-(a*p`2))/p`2>=1 or 1<=(a*p`2)/p`2
                    & 1>=(-(a*p`2))/p`2 by A94,XCMPLX_1:60;
                    (a*p`2)/p`2=a by A94,XCMPLX_1:89;
                    hence 1<=a & -a<=1 or 1>=a & 1<=-a by A106,XCMPLX_1:187;
                  end;
                end;
                then 1<=a & -a<=1 or 1>=a & -1>=--a by XREAL_1:24;
                hence 1<=a or -1>=a;
              end;
              case
A107:           p`1=(-a)*p`2;
                now
                  per cases by A94;
                  case
                    p`2>0;
                    then p`2/p`2<= (-a)*p`2/p`2 & (-((-a)*p`2))/p`2<=p`2/p`2
or p`2/p`2>=((-a)*p`2)/p`2 & p`2/p`2<=(-((-a)*p`2))/p`2 by A82,A107,XREAL_1:72;
                    then 1<= (-a)*p`2/p`2 & (-((-a)*p`2))/p `2<=1 or 1>=((-a)
                    *p`2)/p`2 & 1<=(-((-a)*p`2))/p`2 by A94,XCMPLX_1:60;
                    then
A108:               1<= (-a) & -(((-a)*p`2)/p`2)<=1 or 1>=(-a) & 1<=-(((
                    -a)*p`2)/p`2) by A94,XCMPLX_1:89,187;
                    ((-a)*p`2)/p`2=(-a) by A94,XCMPLX_1:89;
                    hence 1<=a & -a<=1 or 1>=a & 1<=-a by A108;
                  end;
                  case
                    p`2<0;
                    then p`2/p`2>= (-a)*p`2/p`2 & (-((-a)*p`2))/p`2>=p`2/p`2
or p`2/p`2<=((-a)*p`2)/p`2 & p`2/p`2>=(-((-a)*p`2))/p`2 by A82,A107,XREAL_1:73;
                    then 1>= (-a)*p`2/p`2 & (-((-a)*p`2))/p `2>=1 or 1<=((-a)
                    *p`2)/p`2 & 1>=(-((-a)*p`2))/p`2 by A94,XCMPLX_1:60;
                    then
A109:               1>= -a & -(((-a)*p`2)/p`2)>=1 or 1<=-a & 1>=-(((-a)*
                    p`2)/p`2) by A94,XCMPLX_1:89,187;
                    ((-a)*p`2)/p`2=(-a) by A94,XCMPLX_1:89;
                    hence 1<=a & -a<=1 or 1>=a & 1<=-a by A109;
                  end;
                end;
                then 1<=a & -a<=1 or 1>=a & -1>=--a by XREAL_1:24;
                hence 1<=a or -1>=a;
              end;
            end;
A110:       now
              per cases by A103,A97,XXREAL_0:1;
              case
                a=1;
                then (p`1)^2/(p`2)^2=1 by A101,XCMPLX_1:76;
                then
A111:           (p`1)^2=(p`2)^2 by XCMPLX_1:58;
                (p`2/p`1)^2=(p`2)^2/(p`1)^2 by XCMPLX_1:76;
                hence (p`2/p`1)^2=(q`1/q`2)^2 by A101,A111,XCMPLX_1:76;
              end;
              case
                a=-1;
                then (p`1)^2/(p`2)^2=1 by A101,XCMPLX_1:76;
                then
A112:           (p`1)^2=(p`2)^2 by XCMPLX_1:58;
                (p`2/p`1)^2=(p`2)^2/(p`1)^2 by XCMPLX_1:76;
                hence (p`2/p`1)^2=(q`1/q`2)^2 by A101,A112,XCMPLX_1:76;
              end;
            end;
            then p`1=q`1/sqrt(1+(q`1/q`2)^2)*sqrt(1+(q`1/q`2)^2) by A87,A91,
XCMPLX_1:87;
            then
A113:       p`1=q`1 by A77,XCMPLX_1:87;
            p`2=q`2/sqrt(1+(q`1/q`2)^2)*sqrt(1+(q`1/q`2)^2) by A89,A91,A110,
XCMPLX_1:87;
            then p`2=q`2 by A77,XCMPLX_1:87;
            then p=|[q`1,q`2]|by A113,EUCLID:53;
            hence thesis by EUCLID:53;
          end;
        end;
        hence thesis;
      end;
      case
A114:   p<>0.TOP-REAL 2 & not (p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p `2<=-p`1);
        then p`2<>0;
        then
A115:   (p`2)^2>0 by SQUARE_1:12;
A116:   sqrt(1+(p`1/p`2)^2)>0 by Lm1,SQUARE_1:25;
A117:   1+(p`1/p`2)^2>0 by Lm1;
A118:   Sq_Circ.p=|[p`1/sqrt(1+(p`1/p`2)^2),p`2/sqrt(1+(p`1/p`2)^2)]| by A114
,Def1;
        then
A119:   p`1/sqrt(1+(p`1/p`2)^2)=q`1/sqrt(1+(q`1/q`2)^2) by A3,A78,A75,EUCLID:52
;
        then (p`1)^2/(sqrt(1+(p`1/p`2)^2))^2=(q`1/sqrt(1+(q`1/q`2)^2))^2 by
XCMPLX_1:76;
        then (p`1)^2/(sqrt(1+(p`1/p`2)^2))^2=(q`1)^2/(sqrt(1+(q`1/q`2)^2))^2
        by XCMPLX_1:76;
        then (p`1)^2/(1+(p`1/p`2)^2)=(q`1)^2/(sqrt(1+(q`1/q`2)^2))^2 by A117,
SQUARE_1:def 2;
        then
A120:   (p`1)^2/(1+(p`1/p`2)^2)=(q`1)^2/(1+(q`1/q`2)^2) by A76,SQUARE_1:def 2;
A121:   p`2/sqrt(1+(p`1/p`2)^2)=q`2/sqrt(1+(q`1/q`2)^2) by A3,A78,A74,A118,
EUCLID:52;
        then (p`2)^2/(sqrt(1+(p`1/p`2)^2))^2=(q`2/sqrt(1+(q`1/q`2)^2))^2 by
XCMPLX_1:76;
        then (p`2)^2/(sqrt(1+(p`1/p`2)^2))^2=(q`2)^2/(sqrt(1+(q`1/q`2)^2))^2
        by XCMPLX_1:76;
        then (p`2)^2/(1+(p`1/p`2)^2)=(q`2)^2/(sqrt(1+(q`1/q`2)^2))^2 by A117,
SQUARE_1:def 2;
        then (p`2)^2/(1+(p`1/p`2)^2)=(q`2)^2/(1+(q`1/q`2)^2) by A76,
SQUARE_1:def 2;
        then (p`2)^2/(1+(p`1/p`2)^2)/(p`2)^2=(q`2)^2/(p`2)^2/(1+(q`1/q`2)^2)
        by XCMPLX_1:48;
        then (p`2)^2/(p`2)^2/(1+(p`1/p`2)^2)=(q`2)^2/(p`2)^2/(1+(q`1/q`2)^2)
        by XCMPLX_1:48;
        then 1/(1+(p`1/p`2)^2)=(q`2)^2/(p`2)^2/(1+(q`1/q`2)^2) by A115,
XCMPLX_1:60;
        then
A122:   1/(1+(p`1/p`2)^2)*(1+(q`1/q`2)^2)=(q`2)^2/(p`2)^2 by A76,XCMPLX_1:87;
        q`2<>0 by A73;
        then
A123:   (q`2)^2>0 by SQUARE_1:12;
        now
          per cases;
          case
A124:       p`1=0;
            then (q`1)^2=0 by A76,A120,XCMPLX_1:50;
            then
A125:       q`1=0 by XCMPLX_1:6;
            then p=|[0,q`2]|by A3,A78,A118,A124,EUCLID:53;
            hence thesis by A125,EUCLID:53;
          end;
          case
            p`1<>0;
            then
A126:       (p`1)^2>0 by SQUARE_1:12;
            (p`1)^2/(1+(p`1/p`2)^2)/(p`1)^2=(q`1)^2/(p`1)^2/(1+(q`1/q`2)
            ^2) by A120,XCMPLX_1:48;
            then (p`1)^2/(p`1)^2/(1+(p`1/p`2)^2)=(q`1)^2/(p`1)^2/(1+(q`1/q`2)
            ^2) by XCMPLX_1:48;
            then 1/(1+(p`1/p`2)^2)=(q`1)^2/(p`1)^2/(1+(q`1/q`2)^2) by A126,
XCMPLX_1:60;
            then 1/(1+(p`1/p`2)^2)*(1+(q`1/q`2)^2)=(q`1)^2/(p`1)^2 by A76,
XCMPLX_1:87;
            then
(q`2)^2/(q`2)^2/(p`2)^2=(q`1)^2/(p`1)^2/(q`2)^2 by A122,XCMPLX_1:48;
            then 1/(p`2)^2=(q`1)^2/(p`1)^2/(q`2)^2 by A123,XCMPLX_1:60;
            then 1/(p`2)^2*(p`1)^2=(p`1)^2*((q`1)^2/(p`1)^2)/(q`2)^2 by
XCMPLX_1:74;
            then 1/(p`2)^2*(p`1)^2=(q`1)^2/(q`2)^2 by A126,XCMPLX_1:87;
            then (p`1)^2/(p`2)^2=(q`1)^2/(q`2)^2 by XCMPLX_1:99;
            then (p`1/p`2)^2=(q`1)^2/(q`2)^2 by XCMPLX_1:76;
            then
A127:       (1+(p`1/p`2)^2)=(1+(q`1/q`2)^2) by XCMPLX_1:76;
            then p`1=q`1/sqrt(1+(q`1/q`2)^2)*sqrt(1+(q`1/q`2)^2) by A119,A116,
XCMPLX_1:87;
            then
A128:       p`1=q`1 by A77,XCMPLX_1:87;
            p`2=q`2/sqrt(1+(q`1/q`2)^2)*sqrt(1+(q`1/q`2)^2) by A121,A116,A127,
XCMPLX_1:87;
            then p`2=q`2 by A77,XCMPLX_1:87;
            then p=|[q`1,q`2]|by A128,EUCLID:53;
            hence thesis by EUCLID:53;
          end;
        end;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
end;
