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

theorem Th71:
  for cn being Real st -1<cn & cn<1 holds cn-FanMorphN is one-to-one
proof
  let cn be Real;
  assume that
A1: -1<cn and
A2: cn<1;
  for x1,x2 being object st x1 in dom (cn-FanMorphN) & x2 in dom (cn
  -FanMorphN) & (cn-FanMorphN).x1=(cn-FanMorphN).x2 holds x1=x2
  proof
    let x1,x2 be object;
    assume that
A3: x1 in dom (cn-FanMorphN) and
A4: x2 in dom (cn-FanMorphN) and
A5: (cn-FanMorphN).x1=(cn-FanMorphN).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-cn>0 by A2,XREAL_1:149;
    now
      per cases by JGRAPH_2:3;
      case
A7:     q`2<=0;
        then
A8:     (cn-FanMorphN).q=q by Th49;
        now
          per cases by JGRAPH_2:3;
          case
            p`2<=0;
            hence thesis by A5,A8,Th49;
          end;
          case
A9:         p<>0.TOP-REAL 2 & p`1/|.p.|>=cn & p`2>=0;
            set p4= |[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( sqrt(1-((p`1/|.
            p.|-cn)/(1-cn))^2))]|;
A10:        (|.p.|)^2 =(p`1)^2+(p`2)^2 by JGRAPH_3:1;
            0<=(p`2)^2 by XREAL_1:63;
            then 0+(p`1)^2<=(p`1)^2+(p`2)^2 by XREAL_1:7;
            then
A11:        (p`1)^2/(|.p.|)^2 <= (|.p.|)^2/(|.p.|)^2 by A10,XREAL_1:72;
A12:        |.p.|>0 by A9,Lm1;
            then (|.p.|)^2>0 by SQUARE_1:12;
            then (p`1)^2/(|.p.|)^2 <= 1 by A11,XCMPLX_1:60;
            then ((p`1)/|.p.|)^2 <= 1 by XCMPLX_1:76;
            then 1>=p`1/|.p.| by SQUARE_1:51;
            then 1-cn>=p`1/|.p.|-cn by XREAL_1:9;
            then -(1-cn)<= -( p`1/|.p.|-cn) by XREAL_1:24;
            then (-(1-cn))/(1-cn)<=(-( p`1/|.p.|-cn))/(1-cn) by A6,XREAL_1:72;
            then
A13:        -1<=(-( p`1/|.p.|-cn))/(1-cn) by A6,XCMPLX_1:197;
A14:        (p`1/|.p.|-cn)>=0 by A9,XREAL_1:48;
A15:        cn-FanMorphN.p= |[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*(
            sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2))]| by A1,A2,A9,Th51;
            (p`1/|.p.|-cn)>= 0 by A9,XREAL_1:48;
            then ((-(p`1/|.p.|-cn))/(1-cn))^2<=1^2 by A6,A13,SQUARE_1:49;
            then
A16:        1-((-(p`1/|.p.|-cn))/(1-cn))^2>=0 by XREAL_1:48;
            then sqrt(1-((-(p`1/|.p.|-cn))/(1-cn))^2)>=0 by SQUARE_1:def 2;
            then sqrt(1-(-(p`1/|.p.|-cn))^2/(1-cn)^2)>=0 by XCMPLX_1:76;
            then sqrt(1-(p`1/|.p.|-cn)^2/(1-cn)^2)>=0;
            then sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)>=0 by XCMPLX_1:76;
            then p4`2= |.p.|*( sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)) & q`2=0 by A5
,A7,A8,A15,EUCLID:52;
            then
A17:        ( sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2))=0 by A5,A8,A15,A12,XCMPLX_1:6;
            1-(-((p`1/|.p.|-cn))/(1-cn))^2>=0 by A16,XCMPLX_1:187;
            then 1-((p`1/|.p.|-cn)/(1-cn))^2=0 by A17,SQUARE_1:24;
            then 1= (p`1/|.p.|-cn)/(1-cn) by A6,A14,SQUARE_1:18,22;
            then 1 *(1-cn)=(p`1/|.p.|-cn) by A6,XCMPLX_1:87;
            then 1 *|.p.|=p`1 by A12,XCMPLX_1:87;
            then p`2=0 by A10,XCMPLX_1:6;
            hence thesis by A5,A8,Th49;
          end;
          case
A18:        p<>0.TOP-REAL 2 & p`1/|.p.|<cn & p`2>=0;
            then
A19:        |.p.|<>0 by TOPRNS_1:24;
            then
A20:        (|.p.|)^2>0 by SQUARE_1:12;
            set p4= |[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( sqrt(1-((p`1/|.
            p.|-cn)/(1+cn))^2))]|;
A21:        (|.p.|)^2 =(p`1)^2+(p`2)^2 by JGRAPH_3:1;
A22:        1+cn>0 by A1,XREAL_1:148;
A23:        (p`1/|.p.|-cn)<=0 by A18,XREAL_1:47;
            then
A24:        -1<=(-( p`1/|.p.|-cn))/(1+cn) by A22;
A25:        cn-FanMorphN.p= |[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*(
            sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2))]| by A1,A2,A18,Th51;
            0<=(p`2)^2 by XREAL_1:63;
            then 0+(p`1)^2<=(p`1)^2+(p`2)^2 by XREAL_1:7;
            then (p`1)^2/(|.p.|)^2 <= (|.p.|)^2/(|.p.|)^2 by A21,XREAL_1:72;
            then (p`1)^2/(|.p.|)^2 <= 1 by A20,XCMPLX_1:60;
            then ((p`1)/|.p.|)^2 <= 1 by XCMPLX_1:76;
            then (-((p`1)/|.p.|))^2 <= 1;
            then 1>= -p`1/|.p.| by SQUARE_1:51;
            then (1+cn)>= -p`1/|.p.|+cn by XREAL_1:7;
            then (-(p`1/|.p.|-cn))/(1+cn)<=1 by A22,XREAL_1:185;
            then ((-(p`1/|.p.|-cn))/(1+cn))^2<=1^2 by A24,SQUARE_1:49;
            then
A26:        1-((-(p`1/|.p.|-cn))/(1+cn))^2>=0 by XREAL_1:48;
            then sqrt(1-((-(p`1/|.p.|-cn))/(1+cn))^2)>=0 by SQUARE_1:def 2;
            then sqrt(1-(-(p`1/|.p.|-cn))^2/(1+cn)^2)>=0 by XCMPLX_1:76;
            then sqrt(1-((p`1/|.p.|-cn))^2/(1+cn)^2)>=0;
            then sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2)>=0 by XCMPLX_1:76;
            then p4`2= |.p.|*( sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2)) & q`2=0 by A5
,A7,A8,A25,EUCLID:52;
            then
A27:        ( sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2))=0 by A5,A8,A25,A19,XCMPLX_1:6;
            1-(-((p`1/|.p.|-cn))/(1+cn))^2>=0 by A26,XCMPLX_1:187;
            then 1-((p`1/|.p.|-cn)/(1+cn))^2=0 by A27,SQUARE_1:24;
            then 1=sqrt((-((p`1/|.p.|-cn)/(1+cn)))^2);
            then 1= -((p`1/|.p.|-cn)/(1+cn)) by A22,A23,SQUARE_1:22;
            then 1= ((-(p`1/|.p.|-cn))/(1+cn)) by XCMPLX_1:187;
            then 1 *(1+cn)=-(p`1/|.p.|-cn) by A22,XCMPLX_1:87;
            then 1+cn-cn=-p`1/|.p.|;
            then 1=(-p`1)/|.p.| by XCMPLX_1:187;
            then 1 *|.p.|=-p`1 by A18,TOPRNS_1:24,XCMPLX_1:87;
            then (p`1)^2-(p`1)^2 =(p`2)^2 by A21,XCMPLX_1:26;
            then p`2=0 by XCMPLX_1:6;
            hence thesis by A5,A8,Th49;
          end;
        end;
        hence thesis;
      end;
      case
A28:    q`1/|.q.|>=cn & q`2>=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.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( sqrt(1-((q`1/|.q.|-
        cn)/(1-cn))^2))]|;
A30:    q4`1= |.q.|* ((q`1/|.q.|-cn)/(1-cn)) by EUCLID:52;
A31:    cn-FanMorphN.q= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( sqrt(1
        -((q`1/|.q.|-cn)/(1-cn))^2))]| by A1,A2,A28,Th51;
A32:    q4`2= |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)) by EUCLID:52;
        now
          per cases by JGRAPH_2:3;
          case
A33:        p`2<=0;
            then
A34:        (cn-FanMorphN).p=p by Th49;
A35:        |.q.|<>0 by A28,TOPRNS_1:24;
            then
A36:        (|.q.|)^2>0 by SQUARE_1:12;
A37:        (q`1/|.q.|-cn)>= 0 by A28,XREAL_1:48;
A38:        (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1;
A39:        (q`1/|.q.|-cn)>=0 by A28,XREAL_1:48;
A40:        1-cn>0 by A2,XREAL_1:149;
            0<=(q`2)^2 by XREAL_1:63;
            then 0+(q`1)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7;
            then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by A38,XREAL_1:72;
            then (q`1)^2/(|.q.|)^2 <= 1 by A36,XCMPLX_1:60;
            then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76;
            then 1>=q`1/|.q.| by SQUARE_1:51;
            then 1-cn>=q`1/|.q.|-cn by XREAL_1:9;
            then -(1-cn)<= -( q`1/|.q.|-cn) by XREAL_1:24;
            then (-(1-cn))/(1-cn)<=(-( q`1/|.q.|-cn))/(1-cn) by A40,XREAL_1:72;
            then -1<=(-( q`1/|.q.|-cn))/(1-cn) by A40,XCMPLX_1:197;
            then ((-(q`1/|.q.|-cn))/(1-cn))^2<=1^2 by A40,A37,SQUARE_1:49;
            then
A41:        1-((-(q`1/|.q.|-cn))/(1-cn))^2>=0 by XREAL_1:48;
            then sqrt(1-((-(q`1/|.q.|-cn))/(1-cn))^2)>=0 by SQUARE_1:def 2;
            then sqrt(1-(-(q`1/|.q.|-cn))^2/(1-cn)^2)>=0 by XCMPLX_1:76;
            then sqrt(1-(q`1/|.q.|-cn)^2/(1-cn)^2)>=0;
            then sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)>=0 by XCMPLX_1:76;
            then p`2=0 by A5,A31,A33,A34,EUCLID:52;
            then
A42:        ( sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2))=0 by A5,A31,A32,A34,A35,
XCMPLX_1:6;
            1-(-((q`1/|.q.|-cn))/(1-cn))^2>=0 by A41,XCMPLX_1:187;
            then 1-((q`1/|.q.|-cn)/(1-cn))^2=0 by A42,SQUARE_1:24;
            then 1= (q`1/|.q.|-cn)/(1-cn) by A40,A39,SQUARE_1:18,22;
            then 1 *(1-cn)=(q`1/|.q.|-cn) by A40,XCMPLX_1:87;
            then 1 *|.q.|=q`1 by A28,TOPRNS_1:24,XCMPLX_1:87;
            then q`2=0 by A38,XCMPLX_1:6;
            hence thesis by A5,A34,Th49;
          end;
          case
A43:        p<>0.TOP-REAL 2 & p`1/|.p.|>=cn & p`2>=0;
            0<=(q`2)^2 by XREAL_1:63;
            then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`1)^2<=(q`1)^2+(q`2)^2 by
JGRAPH_3:1,XREAL_1:7;
            then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72;
            then (q`1)^2/(|.q.|)^2 <= 1 by A29,XCMPLX_1:60;
            then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76;
            then 1>=q`1/|.q.| by SQUARE_1:51;
            then 1-cn>=q`1/|.q.|-cn by XREAL_1:9;
            then -(1-cn)<= -( q`1/|.q.|-cn) by XREAL_1:24;
            then (-(1-cn))/(1-cn)<=(-( q`1/|.q.|-cn))/(1-cn) by A6,XREAL_1:72;
            then
A44:        -1<=(-( q`1/|.q.|-cn))/(1-cn) by A6,XCMPLX_1:197;
            (q`1/|.q.|-cn)>= 0 by A28,XREAL_1:48;
            then ((-(q`1/|.q.|-cn))/(1-cn))^2<=1^2 by A6,A44,SQUARE_1:49;
            then 1-((-(q`1/|.q.|-cn))/(1-cn))^2>=0 by XREAL_1:48;
            then
A45:        1-(-((q`1/|.q.|-cn))/(1-cn))^2>=0 by XCMPLX_1:187;
            q4`2= |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)) by EUCLID:52;
            then
A46:        (q4`2)^2= (|.q.|)^2*( sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2))^2
              .= (|.q.|)^2*(1-((q`1/|.q.|-cn)/(1-cn))^2) by A45,SQUARE_1:def 2;
A47:        q4`1= |.q.|* ((q`1/|.q.|-cn)/(1-cn)) 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`2)^2 by XREAL_1:63;
            then (|.p.|)^2 =(p`1)^2+(p`2)^2 & 0+(p`1)^2<=(p`1)^2+(p`2)^2 by
JGRAPH_3:1,XREAL_1:7;
            then
A50:        (p`1)^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`1)^2/(|.p.|)^2 <= 1 by A50,XCMPLX_1:60;
            then ((p`1)/|.p.|)^2 <= 1 by XCMPLX_1:76;
            then 1>=p`1/|.p.| by SQUARE_1:51;
            then 1-cn>=p`1/|.p.|-cn by XREAL_1:9;
            then -(1-cn)<= -( p`1/|.p.|-cn) by XREAL_1:24;
            then (-(1-cn))/(1-cn)<=(-( p`1/|.p.|-cn))/(1-cn) by A6,XREAL_1:72;
            then
A51:        -1<=(-( p`1/|.p.|-cn))/(1-cn) by A6,XCMPLX_1:197;
            (p`1/|.p.|-cn)>= 0 by A43,XREAL_1:48;
            then ((-(p`1/|.p.|-cn))/(1-cn))^2<=1^2 by A6,A51,SQUARE_1:49;
            then 1-((-(p`1/|.p.|-cn))/(1-cn))^2>=0 by XREAL_1:48;
            then
A52:        1-(-((p`1/|.p.|-cn))/(1-cn))^2>=0 by XCMPLX_1:187;
            set p4= |[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( sqrt(1-((p`1/|.
            p.|-cn)/(1-cn))^2))]|;
A53:        p4`1= |.p.|* ((p`1/|.p.|-cn)/(1-cn)) by EUCLID:52;
            p4`2= |.p.|*( sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)) by EUCLID:52;
            then
A54:        (p4`2)^2= (|.p.|)^2*( sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2))^2
              .= (|.p.|)^2*(1-((p`1/|.p.|-cn)/(1-cn))^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:        cn-FanMorphN.p= |[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*(
            sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2))]| by A1,A2,A43,Th51;
            then ((p`1/|.p.|-cn)/(1-cn)) =|.q.|* ((q`1/|.q.|-cn)/(1-cn))/|.p
            .| by A5,A31,A30,A43,A53,TOPRNS_1:24,XCMPLX_1:89;
            then (p`1/|.p.|-cn)/(1-cn)=(q`1/|.q.|-cn)/(1-cn) by A5,A31,A43,A57
,A48,A55,TOPRNS_1:24,XCMPLX_1:89;
            then (p`1/|.p.|-cn)/(1-cn)*(1-cn)=q`1/|.q.|-cn by A6,XCMPLX_1:87;
            then p`1/|.p.|-cn=q`1/|.q.|-cn by A6,XCMPLX_1:87;
            then p`1/|.p.|*|.p.|=q`1 by A5,A31,A43,A57,A49,A56,TOPRNS_1:24
,XCMPLX_1:87;
            then
A58:        p`1=q`1 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`2=sqrt((q`2)^2) by A5,A31,A43,A57,A49,A56,A58,SQUARE_1:22;
            then p`2=q`2 by A28,SQUARE_1:22;
            hence thesis by A58,A59,EUCLID:53;
          end;
          case
A60:        p<>0.TOP-REAL 2 & p`1/|.p.|<cn & p`2>=0;
            then p`1/|.p.|-cn<0 by XREAL_1:49;
            then
A61:        ((p`1/|.p.|-cn)/(1+cn))<0 by A1,XREAL_1:141,148;
            set p4= |[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( sqrt(1-((p`1/|.
            p.|-cn)/(1+cn))^2))]|;
A62:        p4`1= |.p.|* ((p`1/|.p.|-cn)/(1+cn)) & q`1/|.q.|-cn>=0 by A28,
EUCLID:52,XREAL_1:48;
A63:        1-cn>0 by A2,XREAL_1:149;
            cn-FanMorphN.p= |[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*(
sqrt(1-((p`1/|.p.| -cn)/(1+cn))^2))]| & |.p.|<>0 by A1,A2,A60,Th51,TOPRNS_1:24;
            hence thesis by A5,A31,A30,A61,A62,A63,XREAL_1:132;
          end;
        end;
        hence thesis;
      end;
      case
A64:    q`1/|.q.|<cn & q`2>=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.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( sqrt(1-((q`1/|.q.|-
        cn)/(1+cn))^2))]|;
A67:    q4`1= |.q.|* ((q`1/|.q.|-cn)/(1+cn)) by EUCLID:52;
A68:    cn-FanMorphN.q= |[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( sqrt(
        1-((q`1/|.q.|-cn)/(1+cn))^2))]| by A1,A2,A64,Th51;
A69:    q4`2= |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)) by EUCLID:52;
        now
          per cases by JGRAPH_2:3;
          case
A70:        p`2<=0;
A71:        (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1;
A72:        1+cn>0 by A1,XREAL_1:148;
            0<=(q`2)^2 by XREAL_1:63;
            then 0+(q`1)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7;
            then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by A71,XREAL_1:72;
            then (q`1)^2/(|.q.|)^2 <= 1 by A66,XCMPLX_1:60;
            then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76;
            then (-((q`1)/|.q.|))^2 <= 1;
            then 1>= -q`1/|.q.| by SQUARE_1:51;
            then (1+cn)>= -q`1/|.q.|+cn by XREAL_1:7;
            then
A73:        (-(q`1/|.q.|-cn))/(1+cn)<=1 by A72,XREAL_1:185;
A74:        (q`1/|.q.|-cn)<=0 by A64,XREAL_1:47;
            then -1<=(-( q`1/|.q.|-cn))/(1+cn) by A72;
            then ((-(q`1/|.q.|-cn))/(1+cn))^2<=1^2 by A73,SQUARE_1:49;
            then
A75:        1-((-(q`1/|.q.|-cn))/(1+cn))^2>=0 by XREAL_1:48;
            then
A76:        1-(-((q`1/|.q.|-cn))/(1+cn))^2>=0 by XCMPLX_1:187;
A77:        (cn-FanMorphN).p=p by A70,Th49;
            sqrt(1-((-(q`1/|.q.|-cn))/(1+cn))^2)>=0 by A75,SQUARE_1:def 2;
            then sqrt(1-(-(q`1/|.q.|-cn))^2/(1+cn)^2)>=0 by XCMPLX_1:76;
            then sqrt(1-((q`1/|.q.|-cn))^2/(1+cn)^2)>=0;
            then sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)>=0 by XCMPLX_1:76;
            then p`2=0 by A5,A68,A70,A77,EUCLID:52;
            then ( sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2))=0 by A5,A68,A69,A65,A77,
XCMPLX_1:6;
            then 1-((q`1/|.q.|-cn)/(1+cn))^2=0 by A76,SQUARE_1:24;
            then 1=sqrt((-((q`1/|.q.|-cn)/(1+cn)))^2);
            then 1= -((q`1/|.q.|-cn)/(1+cn)) by A72,A74,SQUARE_1:22;
            then 1= ((-(q`1/|.q.|-cn))/(1+cn)) by XCMPLX_1:187;
            then 1 *(1+cn)=-(q`1/|.q.|-cn) by A72,XCMPLX_1:87;
            then 1+cn-cn=-q`1/|.q.|;
            then 1=(-q`1)/|.q.| by XCMPLX_1:187;
            then 1 *|.q.|=-q`1 by A64,TOPRNS_1:24,XCMPLX_1:87;
            then (q`1)^2-(q`1)^2 =(q`2)^2 by A71,XCMPLX_1:26;
            then q`2=0 by XCMPLX_1:6;
            hence thesis by A5,A77,Th49;
          end;
          case
A78:        p<>0.TOP-REAL 2 & p`1/|.p.|>=cn & p`2>=0;
            set p4= |[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( sqrt(1-((p`1/|.
            p.|-cn)/(1-cn))^2))]|;
A79:        p4`1= |.p.|* ((p`1/|.p.|-cn)/(1-cn)) & |.q.|<>0 by A64,EUCLID:52
,TOPRNS_1:24;
            q`1/|.q.|-cn<0 by A64,XREAL_1:49;
            then
A80:        ((q`1/|.q.|-cn)/(1+cn))<0 by A1,XREAL_1:141,148;
A81:        1-cn>0 by A2,XREAL_1:149;
            cn-FanMorphN.p= |[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*(
sqrt(1-((p`1/|.p.| -cn)/(1-cn))^2))]| & p`1/|.p.|-cn>=0 by A1,A2,A78,Th51,
XREAL_1:48;
            hence thesis by A5,A68,A67,A80,A79,A81,XREAL_1:132;
          end;
          case
A82:        p<>0.TOP-REAL 2 & p`1/|.p.|<cn & p`2>=0;
            0<=(p`2)^2 by XREAL_1:63;
            then (|.p.|)^2 =(p`1)^2+(p`2)^2 & 0+(p`1)^2<=(p`1)^2+(p`2)^2 by
JGRAPH_3:1,XREAL_1:7;
            then
A83:        (p`1)^2/(|.p.|)^2 <= (|.p.|)^2/(|.p.|)^2 by XREAL_1:72;
A84:        1+cn>0 by A1,XREAL_1:148;
            0<=(q`2)^2 by XREAL_1:63;
            then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`1)^2<=(q`1)^2+(q`2)^2 by
JGRAPH_3:1,XREAL_1:7;
            then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72;
            then (q`1)^2/(|.q.|)^2 <= 1 by A66,XCMPLX_1:60;
            then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76;
            then -1<=q`1/|.q.| by SQUARE_1:51;
            then -1-cn<=q`1/|.q.|-cn by XREAL_1:9;
            then -(-1-cn)>= -(q`1/|.q.|-cn) by XREAL_1:24;
            then
A85:        (-(q`1/|.q.|-cn))/(1+cn)<=1 by A84,XREAL_1:185;
            (q`1/|.q.|-cn)<=0 by A64,XREAL_1:47;
            then -1<=(-( q`1/|.q.|-cn))/(1+cn) by A84;
            then ((-(q`1/|.q.|-cn))/(1+cn))^2<=1^2 by A85,SQUARE_1:49;
            then 1-((-(q`1/|.q.|-cn))/(1+cn))^2>=0 by XREAL_1:48;
            then
A86:        1-(-((q`1/|.q.|-cn))/(1+cn))^2>=0 by XCMPLX_1:187;
            q4`2= |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)) by EUCLID:52;
            then
A87:        (q4`2)^2= (|.q.|)^2*( sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2))^2
              .= (|.q.|)^2*(1-((q`1/|.q.|-cn)/(1+cn))^2) by A86,SQUARE_1:def 2;
A88:        q4`1= |.q.|* ((q`1/|.q.|-cn)/(1+cn)) by EUCLID:52;
            set p4= |[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( sqrt(1-((p`1/|.
            p.|-cn)/(1+cn))^2))]|;
A89:        p4`1= |.p.|* ((p`1/|.p.|-cn)/(1+cn)) by EUCLID:52;
            |.p.|<>0 by A82,TOPRNS_1:24;
            then (|.p.|)^2>0 by SQUARE_1:12;
            then (p`1)^2/(|.p.|)^2 <= 1 by A83,XCMPLX_1:60;
            then ((p`1)/|.p.|)^2 <= 1 by XCMPLX_1:76;
            then -1<=p`1/|.p.| by SQUARE_1:51;
            then -1-cn<=p`1/|.p.|-cn by XREAL_1:9;
            then -(-1-cn)>= -(p`1/|.p.|-cn) by XREAL_1:24;
            then
A90:        (-(p`1/|.p.|-cn))/(1+cn)<=1 by A84,XREAL_1:185;
            (p`1/|.p.|-cn)<=0 by A82,XREAL_1:47;
            then -1<=(-( p`1/|.p.|-cn))/(1+cn) by A84;
            then ((-(p`1/|.p.|-cn))/(1+cn))^2<=1^2 by A90,SQUARE_1:49;
            then 1-((-(p`1/|.p.|-cn))/(1+cn))^2>=0 by XREAL_1:48;
            then
A91:        1-(-((p`1/|.p.|-cn))/(1+cn))^2>=0 by XCMPLX_1:187;
            p4`2= |.p.|*( sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2)) by EUCLID:52;
            then
A92:        (p4`2)^2= (|.p.|)^2*( sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2))^2
              .= (|.p.|)^2*(1-((p`1/|.p.|-cn)/(1+cn))^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:        cn-FanMorphN.p= |[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*(
            sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2))]| by A1,A2,A82,Th51;
            then ((p`1/|.p.|-cn)/(1+cn))=|.q.|* ((q`1/|.q.|-cn)/(1+cn))/|.p.|
            by A5,A68,A67,A82,A89,TOPRNS_1:24,XCMPLX_1:89;
            then (p`1/|.p.|-cn)/(1+cn)=(q`1/|.q.|-cn)/(1+cn) by A5,A68,A82,A97
,A95,A93,TOPRNS_1:24,XCMPLX_1:89;
            then (p`1/|.p.|-cn)/(1+cn)*(1+cn)=q`1/|.q.|-cn by A84,XCMPLX_1:87;
            then p`1/|.p.|-cn=q`1/|.q.|-cn by A84,XCMPLX_1:87;
            then p`1/|.p.|*|.p.|=q`1 by A5,A68,A82,A97,A96,A94,TOPRNS_1:24
,XCMPLX_1:87;
            then
A98:        p`1=q`1 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`2=sqrt((q`2)^2) by A5,A68,A82,A97,A96,A94,A98,SQUARE_1:22;
            then p`2=q`2 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;
