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

theorem
  for f,g being Function of I[01],TOP-REAL 2, C0,KXP,KXN,KYP,KYN being
  Subset of TOP-REAL 2, O,I being Point of I[01] st O=0 & I=1 & f is continuous
one-to-one & g is continuous one-to-one & C0={p: |.p.|<=1}& KXP={q1 where q1 is
Point of TOP-REAL 2: |.q1.|=1 & q1`2<=q1`1 & q1`2>=-q1`1} & KXN={q2 where q2 is
Point of TOP-REAL 2: |.q2.|=1 & q2`2>=q2`1 & q2`2<=-q2`1} & KYP={q3 where q3 is
Point of TOP-REAL 2: |.q3.|=1 & q3`2>=q3`1 & q3`2>=-q3`1} & KYN={q4 where q4 is
Point of TOP-REAL 2: |.q4.|=1 & q4`2<=q4`1 & q4`2<=-q4`1} & f.O in KXN & f.I in
KXP & g.O in KYN & g.I in KYP & rng f c= C0 & rng g c= C0 holds rng f meets rng
  g
proof
A1: dom (Sq_Circ")=the carrier of TOP-REAL 2 by Th29,FUNCT_2:def 1;
  let f,g be Function of I[01],TOP-REAL 2, C0,KXP,KXN,KYP,KYN be Subset of
  TOP-REAL 2, O,I be Point of I[01];
  assume
A2: O=0 & I=1 & f is continuous one-to-one & g is continuous one-to-one
& C0={p: |.p.|<=1}& KXP={q1 where q1 is Point of TOP-REAL 2: |.q1.|=1 & q1`2<=
q1`1 & q1`2>=-q1`1} & KXN={q2 where q2 is Point of TOP-REAL 2: |.q2.|=1 & q2`2
>=q2`1 & q2`2<=-q2`1} & KYP={q3 where q3 is Point of TOP-REAL 2: |.q3.|=1 & q3
  `2>=q3`1 & q3`2>=-q3`1} & KYN={q4 where q4 is Point of TOP-REAL 2: |.q4.|=1 &
q4`2<=q4`1 & q4`2<=-q4`1} & f.O in KXN & f.I in KXP & g.O in KYN & g.I in KYP &
  rng f c= C0 & rng g c= C0;
  then consider p1 being Point of TOP-REAL 2 such that
A3: f.O=p1 and
A4: |.p1.|=1 and
A5: p1`2>=p1`1 and
A6: p1`2<=-p1`1;
  reconsider gg=Sq_Circ"*g as Function of I[01],TOP-REAL 2 by Th29,FUNCT_2:13;
A7: dom g=the carrier of I[01] by FUNCT_2:def 1;
  reconsider ff=Sq_Circ"*f as Function of I[01],TOP-REAL 2 by Th29,FUNCT_2:13;
A8: dom gg=the carrier of I[01] by FUNCT_2:def 1;
A9: dom ff=the carrier of I[01] by FUNCT_2:def 1;
  then
A10: (ff.O)=(Sq_Circ").(f.O) by FUNCT_1:12;
A11: dom f=the carrier of I[01] by FUNCT_2:def 1;
A12: for r being Point of I[01] holds -1<=(ff.r)`1 & (ff.r)`1<=1 & -1<=(gg.
r)`1 & (gg.r)`1<=1 & -1 <=(ff.r)`2 & (ff.r)`2<=1 & -1 <=(gg.r)`2 & (gg.r)`2<=1
  proof
    let r be Point of I[01];
    f.r in rng f by A11,FUNCT_1:3;
    then f.r in C0 by A2;
    then consider p1 being Point of TOP-REAL 2 such that
A13: f.r=p1 and
A14: |.p1.|<=1 by A2;
    g.r in rng g by A7,FUNCT_1:3;
    then g.r in C0 by A2;
    then consider p2 being Point of TOP-REAL 2 such that
A15: g.r=p2 and
A16: |.p2.|<=1 by A2;
A17: (gg.r)=(Sq_Circ").(g.r) by A8,FUNCT_1:12;
A18: now
      per cases;
      case
        p2=0.TOP-REAL 2;
        hence
        -1<=(gg.r)`1 & (gg.r)`1<=1 & -1 <=(gg.r)`2 & (gg.r)`2<=1 by A17,A15
,Th28,JGRAPH_2:3;
      end;
      case
A19:    p2<>0.TOP-REAL 2 & (p2`2<=p2`1 & -p2`1<=p2`2 or p2`2>=p2`1 &
        p2`2<=-p2`1);
        set px=gg.r;
A20:    Sq_Circ".p2=|[p2`1*sqrt(1+(p2`2/p2`1)^2),p2`2*sqrt(1+(p2`2 /p2`1
        )^2 )]| by A19,Th28;
        then
A21:    px`1 = p2`1*sqrt(1+(p2`2/p2`1)^2) by A17,A15,EUCLID:52;
        (|.p2.|)^2<=|.p2.| by A16,SQUARE_1:42;
        then
A22:    (|.p2.|)^2<=1 by A16,XXREAL_0:2;
A23:    (px`2)^2>=0 by XREAL_1:63;
A24:    (px`1)^2 >=0 by XREAL_1:63;
A25:    px`2 = p2`2*sqrt(1+(p2`2/p2`1)^2) by A17,A15,A20,EUCLID:52;
A26:    sqrt(1+(p2`2/p2`1)^2)>0 by Lm1,SQUARE_1:25;
        then p2`2<=p2`1 & -p2`1<=p2`2 or p2`2>=p2`1 & p2`2*sqrt(1+(p2`2/p2`1)
        ^2) <= (-p2`1)*sqrt(1+(p2`2/p2`1)^2) by A19,XREAL_1:64;
        then
A27:    p2`2<=p2`1 & (-p2`1)*sqrt(1+(p2`2/p2`1)^2) <= p2`2*sqrt(1+(p2`2/
        p2`1 )^2) or px`2>=px`1 & px`2<=-px`1 by A21,A25,A26,XREAL_1:64;
        then
A28:    p2`2*sqrt( 1+(p2`2/p2`1)^2) <= p2`1*sqrt(1+(p2`2/p2`1)^2) & -px
        `1<= px `2 or px`2>=px`1 & px`2<=-px`1 by A17,A15,A20,A21,A26,EUCLID:52
,XREAL_1:64;
A29:    now
          assume px`1=0 & px`2=0;
          then p2`1=0 & p2`2=0 by A21,A25,A26,XCMPLX_1:6;
          hence contradiction by A19,EUCLID:53,54;
        end;
        then
A30:    px`1<>0 by A21,A25,A26,A27,XREAL_1:64;
        set q=px;
A31:    (|[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;
A32:    1+(q`2/q`1)^2>0 by Lm1;
A33:    p2=Sq_Circ.px & (|[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 A17,A15,Th43,EUCLID:52,FUNCT_1:32;
        Sq_Circ.q=|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]| by A21
,A25,A29,A28,Def1,JGRAPH_2:3;
        then (|.p2.|)^2= (q`1/sqrt(1+(q`2/q`1)^2))^2+(q`2/sqrt(1+(q`2/q`1)^2)
        ) ^2 by A33,A31,JGRAPH_1:29
          .= (q`1)^2/(sqrt(1+(q`2/q`1)^2))^2+(q`2/sqrt(1+(q`2/q`1)^2))^2 by
XCMPLX_1:76
          .= (q`1)^2/(sqrt(1+(q`2/q`1)^2))^2+(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2
        by XCMPLX_1:76
          .= (q`1)^2/(1+(q`2/q`1)^2)+(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2 by A32,
SQUARE_1:def 2
          .= (q`1)^2/(1+(q`2/q`1)^2)+(q`2)^2/(1+(q`2/q`1)^2) by A32,
SQUARE_1:def 2
          .= ((q`1)^2+(q`2)^2)/(1+(q`2/q`1)^2) by XCMPLX_1:62;
        then ((q`1)^2+(q`2)^2)/(1+(q`2/q`1)^2)*(1+(q`2/q`1)^2)<=1 *(1+(q`2/q
        `1 )^2) by A32,A22,XREAL_1:64;
        then ((q`1)^2+(q`2)^2)<=(1+(q`2/q`1)^2) by A32,XCMPLX_1:87;
        then (px`1)^2+(px`2)^2<=1+(px`2)^2/(px`1)^2 by XCMPLX_1:76;
        then (px`1)^2+(px`2)^2-1<=(px`2)^2/(px`1)^2 by XREAL_1:20;
        then ((px`1)^2+(px`2)^2-1)*(px`1)^2<=(px`2)^2/(px`1)^2*(px`1)^2 by A24,
XREAL_1:64;
        then ((px`1)^2+(px`2)^2-1)*(px`1)^2<=(px`2)^2 by A30,XCMPLX_1:6,87;
        then
A34:    ((px`1)^2-1)*((px`1)^2+(px`2)^2)<=0 by Lm19;
        ((px`1)^2+(px`2)^2)<>0 by A29,COMPLEX1:1;
        then
A35:    ((px`1)^2-1)<=0 by A24,A34,A23,XREAL_1:129;
        then
A36:    px`1>=-1 by SQUARE_1:43;
A37:    px`1<=1 by A35,SQUARE_1:43;
        then q`2<=1 & --q`1>=-q`2 or q`2>=-1 & q`2<=-q`1 by A21,A25,A28,A36,
XREAL_1:24,XXREAL_0:2;
        then q`2<=1 & q`1>=-q`2 or q`2>=-1 & -q`2>=--q`1 by XREAL_1:24;
        then q`2<=1 & 1>=-q`2 or q`2>=-1 & -q`2>=q`1 by A37,XXREAL_0:2;
        then q`2<=1 & -1<=--q`2 or q`2>=-1 & -q`2>=-1 by A36,XREAL_1:24
,XXREAL_0:2;
        hence
        -1<=(gg.r)`1 & (gg.r)`1<=1 & -1 <=(gg.r)`2 & (gg.r)`2<=1 by A35,
SQUARE_1:43,XREAL_1:24;
      end;
      case
A38:    p2<>0.TOP-REAL 2 & not(p2`2<=p2`1 & -p2`1<=p2`2 or p2`2>=p2
        `1 & p2`2<=-p2`1);
        set pz=gg.r;
A39:    Sq_Circ".p2=|[p2`1*sqrt(1+(p2`1/p2`2)^2),p2`2*sqrt(1+(p2`1 /p2`2
        )^2 )]| by A38,Th28;
        then
A40:    pz`2 = p2`2*sqrt(1+(p2`1/p2`2)^2) by A17,A15,EUCLID:52;
A41:    pz`1 = p2`1*sqrt(1+(p2`1/p2`2)^2) by A17,A15,A39,EUCLID:52;
A42:    sqrt(1+(p2`1/p2`2)^2)>0 by Lm1,SQUARE_1:25;
        p2`1<=p2`2 & -p2`2<=p2`1 or p2`1>=p2`2 & p2`1<=-p2`2 by A38,JGRAPH_2:13
;
        then p2`1<=p2`2 & -p2`2<=p2`1 or p2`1>=p2`2 & p2`1*sqrt(1+(p2`1/p2`2)
        ^2) <= (-p2`2)*sqrt(1+(p2`1/p2`2)^2) by A42,XREAL_1:64;
        then
A43:    p2`1<=p2`2 & (-p2`2)*sqrt(1+(p2`1/p2`2)^2) <= p2`1*sqrt(1+(p2`1/
        p2`2) ^2) or pz`1>=pz`2 & pz`1<=-pz`2 by A40,A41,A42,XREAL_1:64;
        then
A44:    p2`1*sqrt(1+(p2`1/p2`2)^2) <= p2`2*sqrt(1+(p2`1/p2`2)^2) & -pz
        `2<=pz `1 or pz`1>=pz`2 & pz`1<=-pz`2 by A17,A15,A39,A40,A42,EUCLID:52
,XREAL_1:64;
A45:    now
          assume that
A46:      pz`2=0 and
          pz`1=0;
          p2`2=0 by A40,A42,A46,XCMPLX_1:6;
          hence contradiction by A38;
        end;
        then
A47:    pz`2<>0 by A40,A41,A42,A43,XREAL_1:64;
A48:    p2=Sq_Circ.pz & (|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/
        pz`2)^2)]|) `2 = pz`2/ sqrt(1+(pz`1/pz`2)^2) by A17,A15,Th43,EUCLID:52
,FUNCT_1:32;
A49:    (pz`2)^2 >=0 by XREAL_1:63;
        (|.p2.|)^2<=|.p2.| by A16,SQUARE_1:42;
        then
A50:    (|.p2.|)^2<=1 by A16,XXREAL_0:2;
A51:    (pz`1)^2>=0 by XREAL_1:63;
A52:    (|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/pz`2)^2)]|)`1 =
        pz`1/ sqrt(1+(pz`1/pz`2)^2) by EUCLID:52;
A53:    1+(pz`1/pz`2)^2>0 by Lm1;
        Sq_Circ.pz=|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/pz`2)
        ^2)]| by A40,A41,A45,A44,Th4,JGRAPH_2:3;
        then (|.p2.|)^2= (pz`2/sqrt(1+(pz`1/pz`2)^2))^2+(pz`1/sqrt(1+(pz`1/pz
        `2)^2))^2 by A48,A52,JGRAPH_1:29
          .= (pz`2)^2/(sqrt(1+(pz`1/pz`2)^2))^2+(pz`1/sqrt(1+(pz`1/pz`2)^2))
        ^2 by XCMPLX_1:76
          .= (pz`2)^2/(sqrt(1+(pz`1/pz`2)^2))^2 +(pz`1)^2/(sqrt(1+(pz`1/pz`2
        )^2))^2 by XCMPLX_1:76
          .= (pz`2)^2/(1+(pz`1/pz`2)^2)+(pz`1)^2/(sqrt(1+(pz`1/pz`2)^2))^2
        by A53,SQUARE_1:def 2
          .= (pz`2)^2/(1+(pz`1/pz`2)^2)+(pz`1)^2/(1+(pz`1/pz`2)^2) by A53,
SQUARE_1:def 2
          .= ((pz`2)^2+(pz`1)^2)/(1+(pz`1/pz`2)^2) by XCMPLX_1:62;
        then ((pz`2)^2+(pz`1)^2)/(1+(pz`1/pz`2)^2)*(1+(pz`1/pz`2)^2) <=1 *(1+
        (pz`1/pz`2)^2) by A53,A50,XREAL_1:64;
        then ((pz`2)^2+(pz`1)^2)<=(1+(pz`1/pz`2)^2) by A53,XCMPLX_1:87;
        then (pz`2)^2+(pz`1)^2<=1+(pz`1)^2/(pz`2)^2 by XCMPLX_1:76;
        then (pz`2)^2+(pz`1)^2-1<=(pz`1)^2/(pz`2)^2 by XREAL_1:20;
        then ((pz`2)^2+(pz`1)^2-1)*(pz`2)^2<=(pz`1)^2/(pz`2)^2*(pz`2)^2 by A49,
XREAL_1:64;
        then ((pz`2)^2+(pz`1)^2-1)*(pz`2)^2<=(pz`1)^2 by A47,XCMPLX_1:6,87;
        then
A54:    ((pz`2)^2-1)*((pz`2)^2+(pz`1)^2)<=0 by Lm19;
        (pz`2)^2+(pz`1)^2<>0 by A45,COMPLEX1:1;
        then
A55:    (pz`2)^2-1<=0 by A49,A54,A51,XREAL_1:129;
        then
A56:    pz`2>=-1 by SQUARE_1:43;
A57:    pz`2<=1 by A55,SQUARE_1:43;
        then pz`1<=1 & --pz`2>=-pz`1 or pz`1>=-1 & pz`1<=-pz`2 by A40,A41,A44
,A56,XREAL_1:24,XXREAL_0:2;
        then pz`1<=1 & 1>=-pz`1 or pz`1>=-1 & -pz`1>=--pz`2 by A57,XREAL_1:24
,XXREAL_0:2;
        then pz`1<=1 & 1>=-pz`1 or pz`1>=-1 & -pz`1>=-1 by A56,XXREAL_0:2;
        then pz`1<=1 & -1<=--pz`1 or pz`1>=-1 & pz`1<=1 by XREAL_1:24;
        hence
        -1<=(gg.r)`1 & (gg.r)`1<=1 & -1 <=(gg.r)`2 & (gg.r)`2<=1 by A55,
SQUARE_1:43;
      end;
    end;
A58: (ff.r)=(Sq_Circ").(f.r) by A9,FUNCT_1:12;
    now
      per cases;
      case
        p1=0.TOP-REAL 2;
        hence
        -1<=(ff.r)`1 & (ff.r)`1<=1 & -1 <=(ff.r)`2 & (ff.r)`2<=1 by A58,A13
,Th28,JGRAPH_2:3;
      end;
      case
A59:    p1<>0.TOP-REAL 2 & (p1`2<=p1`1 & -p1`1<=p1`2 or p1`2>=p1`1 &
        p1`2<=-p1`1);
        set px=ff.r;
        Sq_Circ".p1=|[p1`1*sqrt(1+(p1`2/p1`1)^2),p1`2*sqrt(1+(p1`2 /p1`1
        )^2 ) ]| by A59,Th28;
        then
A60:    px`1 = p1`1*sqrt(1+(p1`2/p1`1)^2) & px`2 = p1`2*sqrt(1+(p1`2/p1
        `1)^2) by A58,A13,EUCLID:52;
A61:    sqrt(1+(p1`2/p1`1)^2)>0 by Lm1,SQUARE_1:25;
        then p1`2<=p1`1 & -p1`1<=p1`2 or p1`2>=p1`1 & p1`2*sqrt(1+(p1`2/p1`1)
        ^2) <= (-p1`1)*sqrt(1+(p1`2/p1`1)^2) by A59,XREAL_1:64;
        then
A62:    p1`2<=p1`1 & (-p1`1)*sqrt(1+(p1`2/p1`1)^2) <= p1`2*sqrt(1+ (p1`2
        /p1`1 )^2) or px`2>=px`1 & px`2<=-px`1 by A60,A61,XREAL_1:64;
        then
A63:    px`2<=px`1 & -px`1<=px`2 or px`2>=px`1 & px`2<=-px`1 by A60,A61,
XREAL_1:64;
A64:    now
          assume px`1=0 & px`2=0;
          then p1`1=0 & p1`2=0 by A60,A61,XCMPLX_1:6;
          hence contradiction by A59,EUCLID:53,54;
        end;
        then
A65:    px`1<>0 by A60,A61,A62,XREAL_1:64;
        (|.p1.|)^2<=|.p1.| by A14,SQUARE_1:42;
        then
A66:    (|.p1.|)^2<=1 by A14,XXREAL_0:2;
A67:    (px`1)^2 >=0 by XREAL_1:63;
A68:    (px`2)^2>=0 by XREAL_1:63;
        set q=px;
A69:    (|[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;
A70:    1+(q`2/q`1)^2>0 by Lm1;
A71:    p1=Sq_Circ.px & (|[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 A58,A13,Th43,EUCLID:52,FUNCT_1:32;
        Sq_Circ.q=|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]| by A64
,A63,Def1,JGRAPH_2:3;
        then (|.p1.|)^2= (q`1/sqrt(1+(q`2/q`1)^2))^2+(q`2/sqrt(1+(q`2/q`1)^2)
        ) ^2 by A71,A69,JGRAPH_1:29
          .= (q`1)^2/(sqrt(1+(q`2/q`1)^2))^2+(q`2/sqrt(1+(q`2/q`1)^2))^2 by
XCMPLX_1:76
          .= (q`1)^2/(sqrt(1+(q`2/q`1)^2))^2+(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2
        by XCMPLX_1:76
          .= (q`1)^2/(1+(q`2/q`1)^2)+(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2 by A70,
SQUARE_1:def 2
          .= (q`1)^2/(1+(q`2/q`1)^2)+(q`2)^2/(1+(q`2/q`1)^2) by A70,
SQUARE_1:def 2
          .= ((q`1)^2+(q`2)^2)/(1+(q`2/q`1)^2) by XCMPLX_1:62;
        then ((q`1)^2+(q`2)^2)/(1+(q`2/q`1)^2)*(1+(q`2/q`1)^2)<=1 *(1+(q`2/q
        `1 )^2) by A70,A66,XREAL_1:64;
        then (q`1)^2+(q`2)^2<=1+(q`2/q`1)^2 by A70,XCMPLX_1:87;
        then (px`1)^2+(px`2)^2<=1+(px`2)^2/(px`1)^2 by XCMPLX_1:76;
        then (px`1)^2+(px`2)^2-1<=(px`2)^2/(px`1)^2 by XREAL_1:20;
        then ((px`1)^2+(px`2)^2-1)*(px`1)^2<=(px`2)^2/(px`1)^2*(px`1)^2 by A67,
XREAL_1:64;
        then ((px`1)^2+(px`2)^2-1)*(px`1)^2<=(px`2)^2 by A65,XCMPLX_1:6,87;
        then
A72:    ((px`1)^2-1)*((px`1)^2+(px`2)^2)<=0 by Lm19;
        ((px`1)^2+(px`2)^2)<>0 by A64,COMPLEX1:1;
        then
A73:    (px`1)^2-1<=0 by A67,A72,A68,XREAL_1:129;
        then
A74:    px`1>=-1 by SQUARE_1:43;
A75:    px`1<=1 by A73,SQUARE_1:43;
        then q`2<=1 & --q`1>=-q`2 or q`2>=-1 & q`2<=-q`1 by A63,A74,XREAL_1:24
,XXREAL_0:2;
        then q`2<=1 & q`1>=-q`2 or q`2>=-1 & -q`2>=--q`1 by XREAL_1:24;
        then q`2<=1 & 1>=-q`2 or q`2>=-1 & -q`2>=q`1 by A75,XXREAL_0:2;
        then q`2<=1 & -1<=--q`2 or q`2>=-1 & -q`2>=-1 by A74,XREAL_1:24
,XXREAL_0:2;
        hence
        -1<=(ff.r)`1 & (ff.r)`1<=1 & -1 <=(ff.r)`2 & (ff.r)`2<=1 by A73,
SQUARE_1:43,XREAL_1:24;
      end;
      case
A76:    p1<>0.TOP-REAL 2 & not(p1`2<=p1`1 & -p1`1<=p1`2 or p1`2>=p1
        `1 & p1`2<=-p1`1);
        set pz=ff.r;
A77:    Sq_Circ".p1=|[p1`1*sqrt(1+(p1`1/p1`2)^2),p1`2*sqrt(1+(p1`1 /p1`2
        ) ^2)]| by A76,Th28;
        then
A78:    pz`2 = p1`2*sqrt(1+(p1`1/p1`2)^2) by A58,A13,EUCLID:52;
A79:    pz`1 = p1`1*sqrt(1+(p1`1/p1`2)^2) by A58,A13,A77,EUCLID:52;
A80:    sqrt(1+(p1`1/p1`2)^2)>0 by Lm1,SQUARE_1:25;
        p1`1<=p1`2 & -p1`2<=p1`1 or p1`1>=p1`2 & p1`1<=-p1`2 by A76,JGRAPH_2:13
;
        then p1`1<=p1`2 & -p1`2<=p1`1 or p1`1>=p1`2 & p1`1*sqrt(1+(p1`1/p1`2)
        ^2) <= (-p1`2)*sqrt(1+(p1`1/p1`2)^2) by A80,XREAL_1:64;
        then
A81:    p1`1<=p1`2 & (-p1`2)*sqrt(1+(p1`1/p1`2)^2) <= p1`1*sqrt(1+(p1`1/
        p1`2 )^2) or pz`1>=pz`2 & pz`1<=-pz`2 by A78,A79,A80,XREAL_1:64;
        then
A82:    p1`1*sqrt(1+(p1`1/p1`2)^2) <= p1`2*sqrt(1+(p1`1/p1`2)^2) & -pz
        `2<=pz `1 or pz`1>=pz`2 & pz`1<=-pz`2 by A58,A13,A77,A78,A80,EUCLID:52
,XREAL_1:64;
A83:    now
          assume that
A84:      pz`2=0 and
          pz`1=0;
          p1`2=0 by A78,A80,A84,XCMPLX_1:6;
          hence contradiction by A76;
        end;
        then
A85:    pz`2<>0 by A78,A79,A80,A81,XREAL_1:64;
A86:    p1=Sq_Circ.pz & (|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/
        pz`2)^2)]|) `2 = pz`2/ sqrt(1+(pz`1/pz`2)^2) by A58,A13,Th43,EUCLID:52
,FUNCT_1:32;
A87:    (pz`2)^2 >=0 by XREAL_1:63;
        (|.p1.|)^2<=|.p1.| by A14,SQUARE_1:42;
        then
A88:    (|.p1.|)^2<=1 by A14,XXREAL_0:2;
A89:    (pz`1)^2>=0 by XREAL_1:63;
A90:    (|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/pz`2)^2)]|)`1 =
        pz`1/ sqrt(1+(pz`1/pz`2)^2) by EUCLID:52;
A91:    1+(pz`1/pz`2)^2>0 by Lm1;
        Sq_Circ.pz=|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/pz`2)
        ^2)]| by A78,A79,A83,A82,Th4,JGRAPH_2:3;
        then (|.p1.|)^2= (pz`2/sqrt(1+(pz`1/pz`2)^2))^2+(pz`1/sqrt(1+(pz`1/pz
        `2)^2))^2 by A86,A90,JGRAPH_1:29
          .= (pz`2)^2/(sqrt(1+(pz`1/pz`2)^2))^2+(pz`1/sqrt(1+(pz`1/pz`2)^2))
        ^2 by XCMPLX_1:76
          .= (pz`2)^2/(sqrt(1+(pz`1/pz`2)^2))^2 +(pz`1)^2/(sqrt(1+(pz`1/pz`2
        )^2))^2 by XCMPLX_1:76
          .= (pz`2)^2/(1+(pz`1/pz`2)^2)+(pz`1)^2/(sqrt(1+(pz`1/pz`2)^2))^2
        by A91,SQUARE_1:def 2
          .= (pz`2)^2/(1+(pz`1/pz`2)^2)+(pz`1)^2/(1+(pz`1/pz`2)^2) by A91,
SQUARE_1:def 2
          .= ((pz`2)^2+(pz`1)^2)/(1+(pz`1/pz`2)^2) by XCMPLX_1:62;
        then ((pz`2)^2+(pz`1)^2)/(1+(pz`1/pz`2)^2)*(1+(pz`1/pz`2)^2) <=1 *(1+
        (pz`1/pz`2)^2) by A91,A88,XREAL_1:64;
        then ((pz`2)^2+(pz`1)^2)<=(1+(pz`1/pz`2)^2) by A91,XCMPLX_1:87;
        then (pz`2)^2+(pz`1)^2<=1+(pz`1)^2/(pz`2)^2 by XCMPLX_1:76;
        then (pz`2)^2+(pz`1)^2-1<=(pz`1)^2/(pz`2)^2 by XREAL_1:20;
        then ((pz`2)^2+(pz`1)^2-1)*(pz`2)^2<=(pz`1)^2/(pz`2)^2*(pz`2)^2 by A87,
XREAL_1:64;
        then ((pz`2)^2+(pz`1)^2-1)*(pz`2)^2<=(pz`1)^2 by A85,XCMPLX_1:6,87;
        then
A92:    ((pz`2)^2-1)*((pz`2)^2+(pz`1)^2)<=0 by Lm19;
        ((pz`2)^2+(pz`1)^2)<>0 by A83,COMPLEX1:1;
        then
A93:    (pz`2)^2-1<=0 by A87,A92,A89,XREAL_1:129;
        then
A94:    pz`2>=-1 by SQUARE_1:43;
A95:    pz`2<=1 by A93,SQUARE_1:43;
        then pz`1<=1 & --pz`2>=-pz`1 or pz`1>=-1 & pz`1<=-pz`2 by A78,A79,A82
,A94,XREAL_1:24,XXREAL_0:2;
        then pz`1<=1 & 1>=-pz`1 or pz`1>=-1 & -pz`1>=--pz`2 by A95,XREAL_1:24
,XXREAL_0:2;
        then pz`1<=1 & 1>=-pz`1 or pz`1>=-1 & -pz`1>=-1 by A94,XXREAL_0:2;
        then pz`1<=1 & -1<=--pz`1 or pz`1>=-1 & pz`1<=1 by XREAL_1:24;
        hence
        -1<=(ff.r)`1 & (ff.r)`1<=1 & -1 <=(ff.r)`2 & (ff.r)`2<=1 by A93,
SQUARE_1:43;
      end;
    end;
    hence thesis by A18;
  end;
  set y = the Element of rng ff /\ rng gg;
A96: p1<>0.TOP-REAL 2 by A4,TOPRNS_1:23;
  then
A97: Sq_Circ".p1=|[p1`1*sqrt(1+(p1`2/p1`1)^2),p1`2*sqrt(1+(p1`2/p1`1)^2 )]|
  by A5,A6,Th28;
  (ff.O)`1=-1 & (ff.I)`1=1 & (gg.O)`2=-1 & (gg.I)`2=1
  proof
    set pz=gg.O;
    set py=ff.I;
    set px=ff.O;
    set q=px;
A98: (|[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;
    set pu=gg.I;
A99: (|[py`1/sqrt(1+(py`2/py`1)^2),py`2/sqrt(1+(py`2/py`1)^2)]|)`1 = py`1/
    sqrt(1+(py`2/py`1)^2) by EUCLID:52;
A100: (|[pu`1/sqrt(1+(pu`1/pu`2)^2),pu`2/sqrt(1+(pu`1/pu`2)^2)]|)`2 = pu`2
    / sqrt(1+(pu`1/pu`2)^2) by EUCLID:52;
A101: 1+(pu`1/pu`2)^2>0 by Lm1;
    Sq_Circ".p1=q by A9,A3,FUNCT_1:12;
    then
A102: p1=Sq_Circ.px by Th43,FUNCT_1:32;
    consider p4 being Point of TOP-REAL 2 such that
A103: g.I=p4 and
A104: |.p4.|=1 and
A105: p4`2>=p4`1 and
A106: p4`2>=-p4`1 by A2;
A107: sqrt(1+(p4`1/p4`2)^2)>0 by Lm1,SQUARE_1:25;
A108: -p4`2<=--p4`1 by A106,XREAL_1:24;
    then
A109: p4`1<=p4`2 & (-p4`2)*sqrt(1+(p4`1/p4`2)^2) <= p4`1*sqrt(1+(p4`1/p4`2
    ) ^2 ) or pu`1>=pu`2 & pu`1<=-pu`2 by A105,A107,XREAL_1:64;
A110: (gg.I)=(Sq_Circ").(g.I) by A8,FUNCT_1:12;
    then
A111: p4=Sq_Circ.pu by A103,Th43,FUNCT_1:32;
A112: p4<>0.TOP-REAL 2 by A104,TOPRNS_1:23;
    then
A113: Sq_Circ".p4=|[p4`1*sqrt(1+(p4`1/p4`2)^2),p4`2*sqrt(1+(p4`1/p4`2) ^2
    )]| by A105,A108,Th30;
    then
A114: pu`2 = p4`2*sqrt(1+(p4`1/p4`2)^2) by A110,A103,EUCLID:52;
A115: pu`1 = p4`1*sqrt(1+(p4`1/p4`2)^2) by A110,A103,A113,EUCLID:52;
A116: now
      assume pu`2=0 & pu`1=0;
      then p4`2=0 & p4`1=0 by A114,A115,A107,XCMPLX_1:6;
      hence contradiction by A112,EUCLID:53,54;
    end;
    p4`1* sqrt(1+(p4`1/p4`2)^2) <= p4`2*sqrt(1+(p4`1/p4`2)^2) & -pu`2<=
pu `1 or pu`1>=pu`2 & pu`1<=-pu`2 by A110,A103,A113,A114,A107,A109,EUCLID:52
,XREAL_1:64;
    then
A117: Sq_Circ.pu=|[pu`1/sqrt(1+(pu`1/pu`2)^2),pu`2/sqrt(1+(pu`1/pu`2)^2)]|
    by A114,A115,A116,Th4,JGRAPH_2:3;
    (|[pu`1/sqrt(1+(pu`1/pu`2)^2),pu`2/sqrt(1+(pu`1/pu`2)^2)]|)`1 = pu`1
    / sqrt(1+(pu`1/pu`2)^2) by EUCLID:52;
    then (|.p4.|)^2= (pu`2/sqrt(1+(pu`1/pu`2)^2))^2+(pu`1/sqrt(1+(pu`1/pu`2)
    ^2 ) ) ^2 by A111,A117,A100,JGRAPH_1:29
      .= (pu`2)^2/(sqrt(1+(pu`1/pu`2)^2))^2+(pu`1/sqrt(1+(pu`1/pu`2)^2))^2
    by XCMPLX_1:76
      .= (pu`2)^2/(sqrt(1+(pu`1/pu`2)^2))^2 +(pu`1)^2/(sqrt(1+(pu`1/pu`2)^2)
    )^2 by XCMPLX_1:76
      .= (pu`2)^2/(1+(pu`1/pu`2)^2)+(pu`1)^2/(sqrt(1+(pu`1/pu`2)^2))^2 by A101,
SQUARE_1:def 2
      .= (pu`2)^2/(1+(pu`1/pu`2)^2)+(pu`1)^2/(1+(pu`1/pu`2)^2) by A101,
SQUARE_1:def 2
      .= ((pu`2)^2+(pu`1)^2)/(1+(pu`1/pu`2)^2) by XCMPLX_1:62;
    then ((pu`2)^2+(pu`1)^2)/(1+(pu`1/pu`2)^2)*(1+(pu`1/pu`2)^2)=1 *(1+(pu`1/
    pu `2)^2) by A104;
    then ((pu`2)^2+(pu`1)^2)=(1+(pu`1/pu`2)^2) by A101,XCMPLX_1:87;
    then
A118: (pu`2)^2+(pu`1)^2-1=(pu`1)^2/(pu`2)^2 by XCMPLX_1:76;
    pu`2<>0 by A114,A115,A107,A116,A109,XREAL_1:64;
    then ((pu`2)^2+(pu`1)^2-1)*(pu`2)^2=(pu`1)^2 by A118,XCMPLX_1:6,87;
    then
A119: ((pu`2)^2-1)*((pu`2)^2+(pu`1)^2)=0;
    ((pu`2)^2+(pu`1)^2)<>0 by A116,COMPLEX1:1;
    then
A120: ((pu`2)^2-1)=0 by A119,XCMPLX_1:6;
A121: sqrt(1+(p1`2/p1`1)^2)>0 by Lm1,SQUARE_1:25;
A122: sqrt(1+(pu`1/pu`2)^2)>0 by Lm1,SQUARE_1:25;
A123: now
      assume
A124: pu`2=-1;
      then -p4`1<0 by A106,A111,A117,A100,A122,XREAL_1:141;
      then --p4`1>-0;
      hence contradiction by A105,A111,A117,A122,A124,EUCLID:52;
    end;
A125: 1+(pz`1/pz`2)^2>0 by Lm1;
A126: px`1 = p1`1*sqrt(1+(p1`2/p1`1)^2) & px`2 = p1`2*sqrt(1+(p1`2/p1`1)^2)
    by A10,A3,A97,EUCLID:52;
A127: now
      assume px`1=0 & px`2=0;
      then p1`1=0 & p1`2=0 by A126,A121,XCMPLX_1:6;
      hence contradiction by A96,EUCLID:53,54;
    end;
    p1`2<=p1`1 & -p1`1<=p1`2 or p1`2>=p1`1 & p1`2*sqrt(1+(p1`2/p1`1)^2)
    <= (-p1`1)*sqrt(1+(p1`2/p1`1)^2) by A5,A6,A121,XREAL_1:64;
    then
A128: p1`2<=p1`1 & (-p1`1)*sqrt(1+(p1`2/p1`1)^2) <= p1`2*sqrt(1+(p1`2/ p1`1
    )^2) or px`2>=px`1 & px`2<=-px`1 by A126,A121,XREAL_1:64;
    then px`2<=px`1 & -px`1<=px`2 or px`2>=px`1 & px`2<=-px`1 by A126,A121,
XREAL_1:64;
    then
A129: Sq_Circ.q=|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]| by A127
,Def1,JGRAPH_2:3;
A130: sqrt(1+(q`2/q`1)^2)>0 by Lm1,SQUARE_1:25;
A131: now
      assume
A132: px`1=1;
      -p1`2>=--p1`1 by A6,XREAL_1:24;
      then -p1`2>0 by A102,A129,A98,A130,A132,XREAL_1:139;
      then --p1`2<-0;
      hence contradiction by A5,A102,A129,A130,A132,EUCLID:52;
    end;
    consider p2 being Point of TOP-REAL 2 such that
A133: f.I=p2 and
A134: |.p2.|=1 and
A135: p2`2<=p2`1 and
A136: p2`2>=-p2`1 by A2;
A137: (ff.I)=(Sq_Circ").(f.I) by A9,FUNCT_1:12;
    then
A138: p2=Sq_Circ.py by A133,Th43,FUNCT_1:32;
A139: p2<>0.TOP-REAL 2 by A134,TOPRNS_1:23;
    then
A140: Sq_Circ".p2=|[p2`1*sqrt(1+(p2`2/p2`1)^2),p2`2*sqrt(1+(p2`2/p2`1) ^2 )
    ]| by A135,A136,Th28;
    then
A141: py`1 = p2`1*sqrt(1+(p2`2/p2`1)^2) by A137,A133,EUCLID:52;
A142: sqrt(1+(p2`2/p2`1)^2)>0 by Lm1,SQUARE_1:25;
A143: py`2 = p2`2*sqrt(1+(p2`2/p2`1)^2) by A137,A133,A140,EUCLID:52;
A144: now
      assume py`1=0 & py`2=0;
      then p2`1=0 & p2`2=0 by A141,A143,A142,XCMPLX_1:6;
      hence contradiction by A139,EUCLID:53,54;
    end;
A145: p2`2<=p2`1 & (-p2`1)*sqrt(1+(p2`2/p2`1)^2) <= p2`2*sqrt(1+(p2`2/p2`1
    ) ^2 ) or py`2>=py`1 & py`2<=-py`1 by A135,A136,A142,XREAL_1:64;
    then p2`2*sqrt(1+(p2`2/p2`1)^2) <= p2`1*sqrt(1+(p2`2/p2`1)^2) & -py `1<=
    py `2 or py`2>=py`1 & py`2<=-py`1 by A137,A133,A140,A141,A142,EUCLID:52
,XREAL_1:64;
    then
A146: Sq_Circ.py=|[py`1/sqrt(1+(py`2/py`1)^2),py`2/sqrt(1+(py`2/py`1)^2)]|
    by A141,A143,A144,Def1,JGRAPH_2:3;
A147: sqrt(1+(py`2/py`1)^2)>0 by Lm1,SQUARE_1:25;
A148: now
      assume
A149: py`1=-1;
      -p2`2<=--p2`1 by A136,XREAL_1:24;
      then -p2`2<0 by A138,A146,A99,A147,A149,XREAL_1:141;
      then --p2`2>-0;
      hence contradiction by A135,A138,A146,A147,A149,EUCLID:52;
    end;
A150: 1+(py`2/py`1)^2>0 by Lm1;
    (|[py`1/sqrt(1+(py`2/py`1)^2),py`2/sqrt(1+(py`2/py`1)^2)]|)`2 = py`2/
    sqrt(1+(py`2/py`1)^2) by EUCLID:52;
    then
    (|.p2.|)^2= (py`1/sqrt(1+(py`2/py`1)^2))^2+(py`2/sqrt(1+(py`2/py`1)^2
    ) ) ^2 by A138,A146,A99,JGRAPH_1:29
      .= (py`1)^2/(sqrt(1+(py`2/py`1)^2))^2+(py`2/sqrt(1+(py`2/py`1)^2))^2
    by XCMPLX_1:76
      .= (py`1)^2/(sqrt(1+(py`2/py`1)^2))^2 +(py`2)^2/(sqrt(1+(py`2/py`1)^2)
    )^2 by XCMPLX_1:76
      .= (py`1)^2/(1+(py`2/py`1)^2)+(py`2)^2/(sqrt(1+(py`2/py`1)^2))^2 by A150,
SQUARE_1:def 2
      .= (py`1)^2/(1+(py`2/py`1)^2)+(py`2)^2/(1+(py`2/py`1)^2) by A150,
SQUARE_1:def 2
      .= ((py`1)^2+(py`2)^2)/(1+(py`2/py`1)^2) by XCMPLX_1:62;
    then
    ((py`1)^2+(py`2)^2)/(1+(py`2/py`1)^2)*(1+(py`2/py`1)^2) =1 *(1+(py`2/
    py`1)^2) by A134;
    then ((py`1)^2+(py`2)^2)=(1+(py`2/py`1)^2) by A150,XCMPLX_1:87;
    then
A151: (py`1)^2+(py`2)^2-1=(py`2)^2/(py`1)^2 by XCMPLX_1:76;
    py`1<>0 by A141,A143,A142,A144,A145,XREAL_1:64;
    then ((py`1)^2+(py`2)^2-1)*(py`1)^2=(py`2)^2 by A151,XCMPLX_1:6,87;
    then
A152: ((py`1)^2-1)*((py`1)^2+(py`2)^2)=0;
    ((py`1)^2+(py`2)^2)<>0 by A144,COMPLEX1:1;
    then
A153: ((py`1)^2-1)=0 by A152,XCMPLX_1:6;
A154: (|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/pz`2)^2)]|)`2 = pz`2/
    sqrt(1+(pz`1/pz`2)^2) by EUCLID:52;
A155: 1+(q`2/q`1)^2>0 by Lm1;
    (|[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;
    then (|.p1.|)^2= (q`1/sqrt(1+(q`2/q`1)^2))^2+(q`2/sqrt(1+(q`2/q`1)^2)) ^2
    by A102,A129,A98,JGRAPH_1:29
      .= (q`1)^2/(sqrt(1+(q`2/q`1)^2))^2+(q`2/sqrt(1+(q`2/q`1)^2))^2 by
XCMPLX_1:76
      .= (q`1)^2/(sqrt(1+(q`2/q`1)^2))^2+(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2 by
XCMPLX_1:76
      .= (q`1)^2/(1+(q`2/q`1)^2)+(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2 by A155,
SQUARE_1:def 2
      .= (q`1)^2/(1+(q`2/q`1)^2)+(q`2)^2/(1+(q`2/q`1)^2) by A155,SQUARE_1:def 2
      .= ((q`1)^2+(q`2)^2)/(1+(q`2/q`1)^2) by XCMPLX_1:62;
    then
    ((q`1)^2+(q`2)^2)/(1+(q`2/q`1)^2)*(1+(q`2/q`1)^2)=1 *(1+(q`2/q`1) ^2)
    by A4;
    then (q`1)^2+(q`2)^2=1+(q`2/q`1)^2 by A155,XCMPLX_1:87;
    then
A156: (px`1)^2+(px`2)^2-1=(px`2)^2/(px`1)^2 by XCMPLX_1:76;
    px`1<>0 by A126,A121,A127,A128,XREAL_1:64;
    then ((px`1)^2+(px`2)^2-1)*(px`1)^2=(px`2)^2 by A156,XCMPLX_1:6,87;
    then
A157: ((px`1)^2-1)*((px`1)^2+(px`2)^2)=0;
    consider p3 being Point of TOP-REAL 2 such that
A158: g.O=p3 and
A159: |.p3.|=1 and
A160: p3`2<=p3`1 and
A161: p3`2<=-p3`1 by A2;
A162: p3<>0.TOP-REAL 2 by A159,TOPRNS_1:23;
A163: (gg.O)=(Sq_Circ").(g.O) by A8,FUNCT_1:12;
    then
A164: p3=Sq_Circ.pz by A158,Th43,FUNCT_1:32;
A165: -p3`2>=--p3`1 by A161,XREAL_1:24;
    then
A166: Sq_Circ".p3=|[p3`1*sqrt(1+(p3`1/p3`2)^2),p3`2*sqrt(1+(p3`1/p3`2) ^2)
    ]| by A160,A162,Th30;
    then
A167: pz`2 = p3`2*sqrt(1+(p3`1/p3`2)^2) by A163,A158,EUCLID:52;
A168: sqrt(1+(p3`1/p3`2)^2)>0 by Lm1,SQUARE_1:25;
A169: pz`1 = p3`1*sqrt(1+(p3`1/p3`2)^2) by A163,A158,A166,EUCLID:52;
A170: now
      assume pz`2=0 & pz`1=0;
      then p3`2=0 & p3`1=0 by A167,A169,A168,XCMPLX_1:6;
      hence contradiction by A162,EUCLID:53,54;
    end;
    p3`1<=p3`2 & -p3`2<=p3`1 or p3`1>=p3`2 & p3`1*sqrt(1+(p3`1/p3`2)^2)
    <= (-p3`2)*sqrt(1+(p3`1/p3`2)^2) by A160,A165,A168,XREAL_1:64;
    then
A171: p3`1<=p3`2 & (-p3`2)*sqrt(1+(p3`1/p3`2)^2) <= p3`1*sqrt(1+(p3`1/p3 `2
    )^2) or pz`1>=pz`2 & pz`1<=-pz`2 by A167,A169,A168,XREAL_1:64;
    then
    p3`1*sqrt(1+(p3`1/p3`2)^2) <= p3`2*sqrt(1+(p3`1/p3`2)^2) & -pz `2<=pz
`1 or pz`1>=pz`2 & pz`1<=-pz`2 by A163,A158,A166,A167,A168,EUCLID:52,XREAL_1:64
    ;
    then
A172: Sq_Circ.pz=|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/pz`2)^2)]|
    by A167,A169,A170,Th4,JGRAPH_2:3;
A173: sqrt(1+(pz`1/pz`2)^2)>0 by Lm1,SQUARE_1:25;
A174: now
      assume
A175: pz`2=1;
      then -p3`1>0 by A161,A164,A172,A154,A173,XREAL_1:139;
      then --p3`1<-0;
      hence contradiction by A160,A164,A172,A173,A175,EUCLID:52;
    end;
    (|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/pz`2)^2)]|)`1 = pz`1/
    sqrt(1+(pz`1/pz`2)^2) by EUCLID:52;
    then (|.p3.|)^2= (pz`2/sqrt(1+(pz`1/pz`2)^2))^2+(pz`1/sqrt(1+(pz`1/pz`2)
    ^2))^2 by A164,A172,A154,JGRAPH_1:29
      .= (pz`2)^2/(sqrt(1+(pz`1/pz`2)^2))^2+(pz`1/sqrt(1+(pz`1/pz`2)^2))^2
    by XCMPLX_1:76
      .= (pz`2)^2/(sqrt(1+(pz`1/pz`2)^2))^2 +(pz`1)^2/(sqrt(1+(pz`1/pz`2)^2)
    )^2 by XCMPLX_1:76
      .= (pz`2)^2/(1+(pz`1/pz`2)^2)+(pz`1)^2/(sqrt(1+(pz`1/pz`2)^2))^2 by A125,
SQUARE_1:def 2
      .= (pz`2)^2/(1+(pz`1/pz`2)^2)+(pz`1)^2/(1+(pz`1/pz`2)^2) by A125,
SQUARE_1:def 2
      .= ((pz`2)^2+(pz`1)^2)/(1+(pz`1/pz`2)^2) by XCMPLX_1:62;
    then ((pz`2)^2+(pz`1)^2)/(1+(pz`1/pz`2)^2)*(1+(pz`1/pz`2)^2)=1 *(1+(pz`1/
    pz `2)^2) by A159;
    then ((pz`2)^2+(pz`1)^2)=(1+(pz`1/pz`2)^2) by A125,XCMPLX_1:87;
    then
A176: (pz`2)^2+(pz`1)^2-1=(pz`1)^2/(pz`2)^2 by XCMPLX_1:76;
    pz`2<>0 by A167,A169,A168,A170,A171,XREAL_1:64;
    then ((pz`2)^2+(pz`1)^2-1)*(pz`2)^2=(pz`1)^2 by A176,XCMPLX_1:6,87;
    then
A177: ((pz`2)^2-1)*((pz`2)^2+(pz`1)^2)=0;
    ((pz`2)^2+(pz`1)^2)<>0 by A170,COMPLEX1:1;
    then
A178: (pz`2)^2-1=0 by A177,XCMPLX_1:6;
    ((px`1)^2+(px`2)^2)<>0 by A127,COMPLEX1:1;
    then ((px`1)^2-1)=0 by A157,XCMPLX_1:6;
    hence thesis by A131,A153,A148,A178,A174,A120,A123,Lm20;
  end;
  then rng ff meets rng gg by A2,A12,Th42,JGRAPH_1:47;
  then
A179: rng ff /\ rng gg <>{};
  then y in rng ff by XBOOLE_0:def 4;
  then consider x1 being object such that
A180: x1 in dom ff and
A181: y=ff.x1 by FUNCT_1:def 3;
  x1 in dom f by A180,FUNCT_1:11;
  then
A182: f.x1 in rng f by FUNCT_1:def 3;
  y in rng gg by A179,XBOOLE_0:def 4;
  then consider x2 being object such that
A183: x2 in dom gg and
A184: y=gg.x2 by FUNCT_1:def 3;
A185: gg.x2=Sq_Circ".(g.x2) by A183,FUNCT_1:12;
  x2 in dom g by A183,FUNCT_1:11;
  then
A186: g.x2 in rng g by FUNCT_1:def 3;
  ff.x1=Sq_Circ".(f.x1) by A180,FUNCT_1:12;
  then f.x1=g.x2 by A181,A184,A1,A182,A186,A185,FUNCT_1:def 4;
  then rng f /\ rng g <> {} by A182,A186,XBOOLE_0:def 4;
  hence thesis;
end;
