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

theorem Th102:
  for sn being Real st -1<sn & sn<1 holds sn-FanMorphE is one-to-one
proof
  let sn be Real;
  assume that
A1: -1<sn and
A2: sn<1;
  for x1,x2 being object st x1 in dom (sn-FanMorphE) & x2 in dom (sn
  -FanMorphE) & (sn-FanMorphE).x1=(sn-FanMorphE).x2 holds x1=x2
  proof
    let x1,x2 be object;
    assume that
A3: x1 in dom (sn-FanMorphE) and
A4: x2 in dom (sn-FanMorphE) and
A5: (sn-FanMorphE).x1=(sn-FanMorphE).x2;
    reconsider p2=x2 as Point of TOP-REAL 2 by A4;
    reconsider p1=x1 as Point of TOP-REAL 2 by A3;
    set q=p1,p=p2;
A6: 1-sn>0 by A2,XREAL_1:149;
    now
      per cases by JGRAPH_2:3;
      case
A7:     q`1<=0;
        then
A8:     (sn-FanMorphE).q=q by Th82;
        now
          per cases by JGRAPH_2:3;
          case
            p`1<=0;
            hence thesis by A5,A8,Th82;
          end;
          case
A9:         p<>0.TOP-REAL 2 & p`2/|.p.|>=sn & p`1>=0;
            then
A10:        (p`2/|.p.|-sn)>=0 by XREAL_1:48;
            set p4= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)), |.p.|* ((p`2
            /|.p.|-sn)/(1-sn))]|;
A11:        (|.p.|)^2 =(p`1)^2+(p`2)^2 by JGRAPH_3:1;
A12:        |.p.|<>0 by A9,TOPRNS_1:24;
            then
A13:        (|.p.|)^2>0 by SQUARE_1:12;
            0<=(p`1)^2 by XREAL_1:63;
            then 0+(p`2)^2<=(p`1)^2+(p`2)^2 by XREAL_1:7;
            then (p`2)^2/(|.p.|)^2 <= (|.p.|)^2/(|.p.|)^2 by A11,XREAL_1:72;
            then (p`2)^2/(|.p.|)^2 <= 1 by A13,XCMPLX_1:60;
            then ((p`2)/|.p.|)^2 <= 1 by XCMPLX_1:76;
            then 1>=p`2/|.p.| by SQUARE_1:51;
            then 1-sn>=p`2/|.p.|-sn by XREAL_1:9;
            then -(1-sn)<= -( p`2/|.p.|-sn) by XREAL_1:24;
            then (-(1-sn))/(1-sn)<=(-( p`2/|.p.|-sn))/(1-sn) by A6,XREAL_1:72;
            then
A14:        -1<=(-( p`2/|.p.|-sn))/(1-sn) by A6,XCMPLX_1:197;
A15:        sn -FanMorphE.p= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2))
            , |.p.|* ((p`2/|.p.|-sn)/(1-sn))]| by A1,A2,A9,Th84;
            (p`2/|.p.|-sn)>= 0 by A9,XREAL_1:48;
            then ((-(p`2/|.p.|-sn))/(1-sn))^2<=1^2 by A6,A14,SQUARE_1:49;
            then
A16:        1-((-(p`2/|.p.|-sn))/(1-sn))^2>=0 by XREAL_1:48;
            then sqrt(1-((-(p`2/|.p.|-sn))/(1-sn))^2)>=0 by SQUARE_1:def 2;
            then sqrt(1-(-(p`2/|.p.|-sn))^2/(1-sn)^2)>=0 by XCMPLX_1:76;
            then sqrt(1-(p`2/|.p.|-sn)^2/(1-sn)^2)>=0;
            then sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)>=0 by XCMPLX_1:76;
            then
            p4`1= |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)) & q`1=0 by A5,A7,A8
,A15,EUCLID:52;
            then
A17:        (sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2))=0 by A5,A8,A15,A12,XCMPLX_1:6;
            1-(-((p`2/|.p.|-sn))/(1-sn))^2>=0 by A16,XCMPLX_1:187;
            then 1-((p`2/|.p.|-sn)/(1-sn))^2=0 by A17,SQUARE_1:24;
            then 1= (p`2/|.p.|-sn)/(1-sn) by A6,A10,SQUARE_1:18,22;
            then 1 *(1-sn)=(p`2/|.p.|-sn) by A6,XCMPLX_1:87;
            then 1 *|.p.|=p`2 by A9,TOPRNS_1:24,XCMPLX_1:87;
            then p`1=0 by A11,XCMPLX_1:6;
            hence thesis by A5,A8,Th82;
          end;
          case
A18:        p<>0.TOP-REAL 2 & p`2/|.p.|<sn & p`1>=0;
            then
A19:        |.p.|<>0 by TOPRNS_1:24;
            then
A20:        (|.p.|)^2>0 by SQUARE_1:12;
            set p4= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)), |.p.|* ((p`2
            /|.p.|-sn)/(1+sn))]|;
A21:        (|.p.|)^2 =(p`1)^2+(p`2)^2 by JGRAPH_3:1;
A22:        1+sn>0 by A1,XREAL_1:148;
A23:        (p`2/|.p.|-sn)<=0 by A18,XREAL_1:47;
            then
A24:        -1<=(-( p`2/|.p.|-sn))/(1+sn) by A22;
A25:        sn -FanMorphE.p= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2))
            , |.p.|* ((p`2/|.p.|-sn)/(1+sn))]| by A1,A2,A18,Th84;
            0<=(p`1)^2 by XREAL_1:63;
            then 0+(p`2)^2<=(p`1)^2+(p`2)^2 by XREAL_1:7;
            then (p`2)^2/(|.p.|)^2 <= (|.p.|)^2/(|.p.|)^2 by A21,XREAL_1:72;
            then (p`2)^2/(|.p.|)^2 <= 1 by A20,XCMPLX_1:60;
            then ((p`2)/|.p.|)^2 <= 1 by XCMPLX_1:76;
            then (-((p`2)/|.p.|))^2 <= 1;
            then 1>= -p`2/|.p.| by SQUARE_1:51;
            then (1+sn)>= -p`2/|.p.|+sn by XREAL_1:7;
            then (-(p`2/|.p.|-sn))/(1+sn)<=1 by A22,XREAL_1:185;
            then ((-(p`2/|.p.|-sn))/(1+sn))^2<=1^2 by A24,SQUARE_1:49;
            then
A26:        1-((-(p`2/|.p.|-sn))/(1+sn))^2>=0 by XREAL_1:48;
            then sqrt(1-((-(p`2/|.p.|-sn))/(1+sn))^2)>=0 by SQUARE_1:def 2;
            then sqrt(1-(-(p`2/|.p.|-sn))^2/(1+sn)^2)>=0 by XCMPLX_1:76;
            then sqrt(1-((p`2/|.p.|-sn))^2/(1+sn)^2)>=0;
            then sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)>=0 by XCMPLX_1:76;
            then
            p4`1= |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)) & q`1=0 by A5,A7,A8
,A25,EUCLID:52;
            then
A27:        (sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2))=0 by A5,A8,A25,A19,XCMPLX_1:6;
            1-(-((p`2/|.p.|-sn))/(1+sn))^2>=0 by A26,XCMPLX_1:187;
            then 1-((p`2/|.p.|-sn)/(1+sn))^2=0 by A27,SQUARE_1:24;
            then 1=sqrt((-((p`2/|.p.|-sn)/(1+sn)))^2);
            then 1= -((p`2/|.p.|-sn)/(1+sn)) by A22,A23,SQUARE_1:22;
            then 1= ((-(p`2/|.p.|-sn))/(1+sn)) by XCMPLX_1:187;
            then 1 *(1+sn)=-(p`2/|.p.|-sn) by A22,XCMPLX_1:87;
            then 1+sn-sn=-p`2/|.p.|;
            then 1=(-p`2)/|.p.| by XCMPLX_1:187;
            then 1 *|.p.|=-p`2 by A18,TOPRNS_1:24,XCMPLX_1:87;
            then (p`2)^2-(p`2)^2 =(p`1)^2 by A21,XCMPLX_1:26;
            then p`1=0 by XCMPLX_1:6;
            hence thesis by A5,A8,Th82;
          end;
        end;
        hence thesis;
      end;
      case
A28:    q`2/|.q.|>=sn & q`1>=0 & q<>0.TOP-REAL 2;
        then |.q.|<>0 by TOPRNS_1:24;
        then
A29:    (|.q.|)^2>0 by SQUARE_1:12;
        set q4= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|* ((q`2/|.q
        .|-sn)/(1-sn))]|;
A30:    q4`2= |.q.|* ((q`2/|.q.|-sn)/(1-sn)) by EUCLID:52;
A31:    sn-FanMorphE.q= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2 )), |.
        q.|* ((q`2/|.q.|-sn)/(1-sn))]| by A1,A2,A28,Th84;
A32:    q4`1= |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)) by EUCLID:52;
        now
          per cases by JGRAPH_2:3;
          case
A33:        p`1<=0;
            then
A34:        (sn-FanMorphE).p=p by Th82;
A35:        |.q.|<>0 by A28,TOPRNS_1:24;
            then
A36:        (|.q.|)^2>0 by SQUARE_1:12;
A37:        (q`2/|.q.|-sn)>= 0 by A28,XREAL_1:48;
A38:        (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1;
A39:        (q`2/|.q.|-sn)>=0 by A28,XREAL_1:48;
A40:        1-sn>0 by A2,XREAL_1:149;
            0<=(q`1)^2 by XREAL_1:63;
            then 0+(q`2)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7;
            then (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by A38,XREAL_1:72;
            then (q`2)^2/(|.q.|)^2 <= 1 by A36,XCMPLX_1:60;
            then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76;
            then 1>=q`2/|.q.| by SQUARE_1:51;
            then 1-sn>=q`2/|.q.|-sn by XREAL_1:9;
            then -(1-sn)<= -( q`2/|.q.|-sn) by XREAL_1:24;
            then (-(1-sn))/(1-sn)<=(-( q`2/|.q.|-sn))/(1-sn) by A40,XREAL_1:72;
            then -1<=(-( q`2/|.q.|-sn))/(1-sn) by A40,XCMPLX_1:197;
            then ((-(q`2/|.q.|-sn))/(1-sn))^2<=1^2 by A40,A37,SQUARE_1:49;
            then
A41:        1-((-(q`2/|.q.|-sn))/(1-sn))^2>=0 by XREAL_1:48;
            then sqrt(1-((-(q`2/|.q.|-sn))/(1-sn))^2)>=0 by SQUARE_1:def 2;
            then sqrt(1-(-(q`2/|.q.|-sn))^2/(1-sn)^2)>=0 by XCMPLX_1:76;
            then sqrt(1-(q`2/|.q.|-sn)^2/(1-sn)^2)>=0;
            then sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)>=0 by XCMPLX_1:76;
            then p`1=0 by A5,A31,A33,A34,EUCLID:52;
            then
A42:        (sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2))=0 by A5,A31,A32,A34,A35,
XCMPLX_1:6;
            1-(-((q`2/|.q.|-sn))/(1-sn))^2>=0 by A41,XCMPLX_1:187;
            then 1-((q`2/|.q.|-sn)/(1-sn))^2=0 by A42,SQUARE_1:24;
            then 1= (q`2/|.q.|-sn)/(1-sn) by A40,A39,SQUARE_1:18,22;
            then 1 *(1-sn)=(q`2/|.q.|-sn) by A40,XCMPLX_1:87;
            then 1 *|.q.|=q`2 by A28,TOPRNS_1:24,XCMPLX_1:87;
            then q`1=0 by A38,XCMPLX_1:6;
            hence thesis by A5,A34,Th82;
          end;
          case
A43:        p<>0.TOP-REAL 2 & p`2/|.p.|>=sn & p`1>=0;
            0<=(q`1)^2 by XREAL_1:63;
            then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`2)^2<=(q`1)^2+(q`2)^2 by
JGRAPH_3:1,XREAL_1:7;
            then (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72;
            then (q`2)^2/(|.q.|)^2 <= 1 by A29,XCMPLX_1:60;
            then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76;
            then 1>=q`2/|.q.| by SQUARE_1:51;
            then 1-sn>=q`2/|.q.|-sn by XREAL_1:9;
            then -(1-sn)<= -( q`2/|.q.|-sn) by XREAL_1:24;
            then (-(1-sn))/(1-sn)<=(-( q`2/|.q.|-sn))/(1-sn) by A6,XREAL_1:72;
            then
A44:        -1<=(-( q`2/|.q.|-sn))/(1-sn) by A6,XCMPLX_1:197;
            (q`2/|.q.|-sn)>= 0 by A28,XREAL_1:48;
            then ((-(q`2/|.q.|-sn))/(1-sn))^2<=1^2 by A6,A44,SQUARE_1:49;
            then 1-((-(q`2/|.q.|-sn))/(1-sn))^2>=0 by XREAL_1:48;
            then
A45:        1-(-((q`2/|.q.|-sn))/(1-sn))^2>=0 by XCMPLX_1:187;
            q4`1= |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)) by EUCLID:52;
            then
A46:        (q4`1)^2= (|.q.|)^2*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2))^2
              .= (|.q.|)^2*(1-((q`2/|.q.|-sn)/(1-sn))^2) by A45,SQUARE_1:def 2;
A47:        q4`2= |.q.|* ((q`2/|.q.|-sn)/(1-sn)) by EUCLID:52;
            (|.q4.|)^2=(q4`1)^2+(q4`2)^2 by JGRAPH_3:1
              .=(|.q.|)^2 by A47,A46;
            then
A48:        sqrt((|.q4.|)^2)=|.q.| by SQUARE_1:22;
            then
A49:        |.q4.|=|.q.| by SQUARE_1:22;
            0<=(p`1)^2 by XREAL_1:63;
            then (|.p.|)^2 =(p`1)^2+(p`2)^2 & 0+(p`2)^2<=(p`1)^2+(p`2)^2 by
JGRAPH_3:1,XREAL_1:7;
            then
A50:        (p`2)^2/(|.p.|)^2 <= (|.p.|)^2/(|.p.|)^2 by XREAL_1:72;
            |.p.|<>0 by A43,TOPRNS_1:24;
            then (|.p.|)^2>0 by SQUARE_1:12;
            then (p`2)^2/(|.p.|)^2 <= 1 by A50,XCMPLX_1:60;
            then ((p`2)/|.p.|)^2 <= 1 by XCMPLX_1:76;
            then 1>=p`2/|.p.| by SQUARE_1:51;
            then 1-sn>=p`2/|.p.|-sn by XREAL_1:9;
            then -(1-sn)<= -( p`2/|.p.|-sn) by XREAL_1:24;
            then (-(1-sn))/(1-sn)<=(-( p`2/|.p.|-sn))/(1-sn) by A6,XREAL_1:72;
            then
A51:        -1<=(-( p`2/|.p.|-sn))/(1-sn) by A6,XCMPLX_1:197;
            (p`2/|.p.|-sn)>= 0 by A43,XREAL_1:48;
            then ((-(p`2/|.p.|-sn))/(1-sn))^2<=1^2 by A6,A51,SQUARE_1:49;
            then 1-((-(p`2/|.p.|-sn))/(1-sn))^2>=0 by XREAL_1:48;
            then
A52:        1-(-((p`2/|.p.|-sn))/(1-sn))^2>=0 by XCMPLX_1:187;
            set p4= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)), |.p.|* ((p`2
            /|.p.|-sn)/(1-sn))]|;
A53:        p4`2= |.p.|* ((p`2/|.p.|-sn)/(1-sn)) by EUCLID:52;
            p4`1= |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)) by EUCLID:52;
            then
A54:        (p4`1)^2= (|.p.|)^2*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2))^2
              .= (|.p.|)^2*(1-((p`2/|.p.|-sn)/(1-sn))^2) by A52,SQUARE_1:def 2;
            (|.p4.|)^2=(p4`1)^2+(p4`2)^2 by JGRAPH_3:1
              .=(|.p.|)^2 by A53,A54;
            then
A55:        sqrt((|.p4.|)^2)=|.p.| by SQUARE_1:22;
            then
A56:        |.p4.|=|.p.| by SQUARE_1:22;
A57:        sn -FanMorphE.p= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)
            ), |.p.|* ((p`2/|.p.|-sn)/(1-sn))]| by A1,A2,A43,Th84;
            then ((p`2/|.p.|-sn)/(1-sn)) =|.q.|* ((q`2/|.q.|-sn)/(1-sn))/|.p
            .| by A5,A31,A30,A43,A53,TOPRNS_1:24,XCMPLX_1:89;
            then (p`2/|.p.|-sn)/(1-sn)=(q`2/|.q.|-sn)/(1-sn) by A5,A31,A43,A57
,A48,A55,TOPRNS_1:24,XCMPLX_1:89;
            then (p`2/|.p.|-sn)/(1-sn)*(1-sn)=q`2/|.q.|-sn by A6,XCMPLX_1:87;
            then p`2/|.p.|-sn=q`2/|.q.|-sn by A6,XCMPLX_1:87;
            then p`2/|.p.|*|.p.|=q`2 by A5,A31,A43,A57,A49,A56,TOPRNS_1:24
,XCMPLX_1:87;
            then
A58:        p`2=q`2 by A43,TOPRNS_1:24,XCMPLX_1:87;
A59:        p=|[p`1,p`2]| by EUCLID:53;
            |.p.|^2=(p`1)^2+(p`2)^2 & |.q.|^2=(q`1)^2+(q`2)^2 by JGRAPH_3:1;
            then p`1=sqrt((q`1)^2) by A5,A31,A43,A57,A49,A56,A58,SQUARE_1:22;
            then p`1=q`1 by A28,SQUARE_1:22;
            hence thesis by A58,A59,EUCLID:53;
          end;
          case
A60:        p<>0.TOP-REAL 2 & p`2/|.p.|<sn & p`1>=0;
            then p`2/|.p.|-sn<0 by XREAL_1:49;
            then
A61:        ((p`2/|.p.|-sn)/(1+sn))<0 by A1,XREAL_1:141,148;
            set p4= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)), |.p.|* ((p`2
            /|.p.|-sn)/(1+sn))]|;
A62:        p4`2= |.p.|* ((p`2/|.p.|-sn)/(1+sn)) & q`2/|.q.|-sn>=0 by A28,
EUCLID:52,XREAL_1:48;
A63:        1-sn>0 by A2,XREAL_1:149;
            sn -FanMorphE.p= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)
), |.p.|* ((p`2/ |.p.|-sn)/(1+sn))]| & |.p.|<>0 by A1,A2,A60,Th84,TOPRNS_1:24;
            hence thesis by A5,A31,A30,A61,A62,A63,XREAL_1:132;
          end;
        end;
        hence thesis;
      end;
      case
A64:    q`2/|.q.|<sn & q`1>=0 & q<>0.TOP-REAL 2;
        then
A65:    |.q.|<>0 by TOPRNS_1:24;
        then
A66:    (|.q.|)^2>0 by SQUARE_1:12;
        set q4= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)), |.q.|* ((q`2/|.q
        .|-sn)/(1+sn))]|;
A67:    q4`2= |.q.|* ((q`2/|.q.|-sn)/(1+sn)) by EUCLID:52;
A68:    sn-FanMorphE.q= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2 )),
        |.q.|* ((q`2/|.q.|-sn)/(1+sn))]| by A1,A2,A64,Th84;
A69:    q4`1= |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)) by EUCLID:52;
        now
          per cases by JGRAPH_2:3;
          case
A70:        p`1<=0;
A71:        (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1;
A72:        1+sn>0 by A1,XREAL_1:148;
            0<=(q`1)^2 by XREAL_1:63;
            then 0+(q`2)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7;
            then (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by A71,XREAL_1:72;
            then (q`2)^2/(|.q.|)^2 <= 1 by A66,XCMPLX_1:60;
            then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76;
            then (-((q`2)/|.q.|))^2 <= 1;
            then 1>= -q`2/|.q.| by SQUARE_1:51;
            then (1+sn)>= -q`2/|.q.|+sn by XREAL_1:7;
            then
A73:        (-(q`2/|.q.|-sn))/(1+sn)<=1 by A72,XREAL_1:185;
A74:        (q`2/|.q.|-sn)<=0 by A64,XREAL_1:47;
            then -1<=(-( q`2/|.q.|-sn))/(1+sn) by A72;
            then ((-(q`2/|.q.|-sn))/(1+sn))^2<=1^2 by A73,SQUARE_1:49;
            then
A75:        1-((-(q`2/|.q.|-sn))/(1+sn))^2>=0 by XREAL_1:48;
            then
A76:        1-(-((q`2/|.q.|-sn))/(1+sn))^2>=0 by XCMPLX_1:187;
A77:        (sn-FanMorphE).p=p by A70,Th82;
            sqrt(1-((-(q`2/|.q.|-sn))/(1+sn))^2)>=0 by A75,SQUARE_1:def 2;
            then sqrt(1-(-(q`2/|.q.|-sn))^2/(1+sn)^2)>=0 by XCMPLX_1:76;
            then sqrt(1-((q`2/|.q.|-sn))^2/(1+sn)^2)>=0;
            then sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)>=0 by XCMPLX_1:76;
            then p`1=0 by A5,A68,A70,A77,EUCLID:52;
            then (sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2))=0 by A5,A68,A69,A65,A77,
XCMPLX_1:6;
            then 1-((q`2/|.q.|-sn)/(1+sn))^2=0 by A76,SQUARE_1:24;
            then 1=sqrt((-((q`2/|.q.|-sn)/(1+sn)))^2);
            then 1= -((q`2/|.q.|-sn)/(1+sn)) by A72,A74,SQUARE_1:22;
            then 1= ((-(q`2/|.q.|-sn))/(1+sn)) by XCMPLX_1:187;
            then 1 *(1+sn)=-(q`2/|.q.|-sn) by A72,XCMPLX_1:87;
            then 1+sn-sn=-q`2/|.q.|;
            then 1=(-q`2)/|.q.| by XCMPLX_1:187;
            then 1 *|.q.|=-q`2 by A64,TOPRNS_1:24,XCMPLX_1:87;
            then (q`2)^2-(q`2)^2 =(q`1)^2 by A71,XCMPLX_1:26;
            then q`1=0 by XCMPLX_1:6;
            hence thesis by A5,A77,Th82;
          end;
          case
A78:        p<>0.TOP-REAL 2 & p`2/|.p.|>=sn & p`1>=0;
            set p4= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)), |.p.|* ((p`2
            /|.p.|-sn)/(1-sn))]|;
A79:        p4`2= |.p.|* ((p`2/|.p.|-sn)/(1-sn)) & |.q.|<>0 by A64,EUCLID:52
,TOPRNS_1:24;
            q`2/|.q.|-sn<0 by A64,XREAL_1:49;
            then
A80:        ((q`2/|.q.|-sn)/(1+sn))<0 by A1,XREAL_1:141,148;
A81:        1-sn>0 by A2,XREAL_1:149;
            sn -FanMorphE.p= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)
), |.p.|* ((p`2/ |.p.|-sn)/(1-sn))]| & p`2/|.p.|-sn>=0 by A1,A2,A78,Th84,
XREAL_1:48;
            hence thesis by A5,A68,A67,A80,A79,A81,XREAL_1:132;
          end;
          case
A82:        p<>0.TOP-REAL 2 & p`2/|.p.|<sn & p`1>=0;
            0<=(p`1)^2 by XREAL_1:63;
            then (|.p.|)^2 =(p`1)^2+(p`2)^2 & 0+(p`2)^2<=(p`1)^2+(p`2)^2 by
JGRAPH_3:1,XREAL_1:7;
            then
A83:        (p`2)^2/(|.p.|)^2 <= (|.p.|)^2/(|.p.|)^2 by XREAL_1:72;
A84:        1+sn>0 by A1,XREAL_1:148;
            0<=(q`1)^2 by XREAL_1:63;
            then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`2)^2<=(q`1)^2+(q`2)^2 by
JGRAPH_3:1,XREAL_1:7;
            then (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72;
            then (q`2)^2/(|.q.|)^2 <= 1 by A66,XCMPLX_1:60;
            then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76;
            then -1<=q`2/|.q.| by SQUARE_1:51;
            then -1-sn<=q`2/|.q.|-sn by XREAL_1:9;
            then -(-1-sn)>= -(q`2/|.q.|-sn) by XREAL_1:24;
            then
A85:        (-(q`2/|.q.|-sn))/(1+sn)<=1 by A84,XREAL_1:185;
            (q`2/|.q.|-sn)<=0 by A64,XREAL_1:47;
            then -1<=(-( q`2/|.q.|-sn))/(1+sn) by A84;
            then ((-(q`2/|.q.|-sn))/(1+sn))^2<=1^2 by A85,SQUARE_1:49;
            then 1-((-(q`2/|.q.|-sn))/(1+sn))^2>=0 by XREAL_1:48;
            then
A86:        1-(-((q`2/|.q.|-sn))/(1+sn))^2>=0 by XCMPLX_1:187;
            q4`1= |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)) by EUCLID:52;
            then
A87:        (q4`1)^2= (|.q.|)^2*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2))^2
              .= (|.q.|)^2*(1-((q`2/|.q.|-sn)/(1+sn))^2) by A86,SQUARE_1:def 2;
A88:        q4`2= |.q.|* ((q`2/|.q.|-sn)/(1+sn)) by EUCLID:52;
            set p4= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)), |.p.|* ((p`2
            /|.p.|-sn)/(1+sn))]|;
A89:        p4`2= |.p.|* ((p`2/|.p.|-sn)/(1+sn)) by EUCLID:52;
            |.p.|<>0 by A82,TOPRNS_1:24;
            then (|.p.|)^2>0 by SQUARE_1:12;
            then (p`2)^2/(|.p.|)^2 <= 1 by A83,XCMPLX_1:60;
            then ((p`2)/|.p.|)^2 <= 1 by XCMPLX_1:76;
            then -1<=p`2/|.p.| by SQUARE_1:51;
            then -1-sn<=p`2/|.p.|-sn by XREAL_1:9;
            then -(-1-sn)>= -(p`2/|.p.|-sn) by XREAL_1:24;
            then
A90:        (-(p`2/|.p.|-sn))/(1+sn)<=1 by A84,XREAL_1:185;
            (p`2/|.p.|-sn)<=0 by A82,XREAL_1:47;
            then -1<=(-( p`2/|.p.|-sn))/(1+sn) by A84;
            then ((-(p`2/|.p.|-sn))/(1+sn))^2<=1^2 by A90,SQUARE_1:49;
            then 1-((-(p`2/|.p.|-sn))/(1+sn))^2>=0 by XREAL_1:48;
            then
A91:        1-(-((p`2/|.p.|-sn))/(1+sn))^2>=0 by XCMPLX_1:187;
            p4`1= |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)) by EUCLID:52;
            then
A92:        (p4`1)^2= (|.p.|)^2*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2))^2
              .= (|.p.|)^2*(1-((p`2/|.p.|-sn)/(1+sn))^2) by A91,SQUARE_1:def 2;
            (|.p4.|)^2=(p4`1)^2+(p4`2)^2 by JGRAPH_3:1
              .=(|.p.|)^2 by A89,A92;
            then
A93:        sqrt((|.p4.|)^2)=|.p.| by SQUARE_1:22;
            then
A94:        |.p4.|=|.p.| by SQUARE_1:22;
            (|.q4.|)^2=(q4`1)^2+(q4`2)^2 by JGRAPH_3:1
              .=(|.q.|)^2 by A88,A87;
            then
A95:        sqrt((|.q4.|)^2)=|.q.| by SQUARE_1:22;
            then
A96:        |.q4.|=|.q.| by SQUARE_1:22;
A97:        sn -FanMorphE.p= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)
            ), |.p.|* ((p`2/|.p.|-sn)/(1+sn))]| by A1,A2,A82,Th84;
            then ((p`2/|.p.|-sn)/(1+sn)) =|.q.|* ((q`2/|.q.|-sn)/(1+sn))/|.p
            .| by A5,A68,A67,A82,A89,TOPRNS_1:24,XCMPLX_1:89;
            then (p`2/|.p.|-sn)/(1+sn)=(q`2/|.q.|-sn)/(1+sn) by A5,A68,A82,A97
,A95,A93,TOPRNS_1:24,XCMPLX_1:89;
            then (p`2/|.p.|-sn)/(1+sn)*(1+sn)=q`2/|.q.|-sn by A84,XCMPLX_1:87;
            then p`2/|.p.|-sn=q`2/|.q.|-sn by A84,XCMPLX_1:87;
            then p`2/|.p.|*|.p.|=q`2 by A5,A68,A82,A97,A96,A94,TOPRNS_1:24
,XCMPLX_1:87;
            then
A98:        p`2=q`2 by A82,TOPRNS_1:24,XCMPLX_1:87;
A99:        p=|[p`1,p`2]| by EUCLID:53;
            |.p.|^2=(p`1)^2+(p`2)^2 & |.q.|^2=(q`1)^2+(q`2)^2 by JGRAPH_3:1;
            then p`1=sqrt((q`1)^2) by A5,A68,A82,A97,A96,A94,A98,SQUARE_1:22;
            then p`1=q`1 by A64,SQUARE_1:22;
            hence thesis by A98,A99,EUCLID:53;
          end;
        end;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
  hence thesis by FUNCT_1:def 4;
end;
