let cn be Real; :: thesis: for p being Point of (TOP-REAL 2) holds |.((cn -FanMorphN ) . p).| = |.p.|
let p be Point of (TOP-REAL 2); :: thesis: |.((cn -FanMorphN ) . p).| = |.p.|
set f = cn -FanMorphN ;
set z = (cn -FanMorphN ) . p;
reconsider q = p as Point of (TOP-REAL 2) ;
reconsider qz = (cn -FanMorphN ) . p as Point of (TOP-REAL 2) ;

per cases ( ( (q `1 ) / |.q.| >= cn & q `2 > 0 ) or ( (q `1 ) / |.q.| < cn & q `2 > 0 ) or q `2 <= 0 ) ;

suppose A1: ( (q `1 ) / |.q.| >= cn & q `2 > 0 ) ; :: thesis: |.((cn -FanMorphN ) . p).| = |.p.|

then A2: (cn -FanMorphN ) . q = |[(|.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))))]| by Th56;
then A3: qz `2 = |.q.| * (sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))) by EUCLID:56;
A4: qz `1 = |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 - cn)) by A2, EUCLID:56;
A5: ((q `1 ) / |.q.|) - cn >= 0 by A1, XREAL_1:50;
A6: |.q.| ^2 = ((q `1 ) ^2 ) + ((q `2 ) ^2 ) by JGRAPH_3:10;
|.q.| <> 0 by A1, JGRAPH_2:11, TOPRNS_1:25;
then A7: |.q.| ^2 > 0 by SQUARE_1:74;
0 <= (q `2 ) ^2 by XREAL_1:65;
then 0 + ((q `1 ) ^2 ) <= ((q `1 ) ^2 ) + ((q `2 ) ^2 ) by XREAL_1:9;
then ((q `1 ) ^2 ) / (|.q.| ^2 ) <= (|.q.| ^2 ) / (|.q.| ^2 ) by A6, XREAL_1:74;
then ((q `1 ) ^2 ) / (|.q.| ^2 ) <= 1 by A7, XCMPLX_1:60;
then ((q `1 ) / |.q.|) ^2 <= 1 by XCMPLX_1:77;
then 1 >= (q `1 ) / |.q.| by SQUARE_1:121;
then A8: 1 - cn >= ((q `1 ) / |.q.|) - cn by XREAL_1:11;

per cases ( 1 - cn = 0 or 1 - cn <> 0 ) ;

suppose A9: 1 - cn = 0 ; :: thesis: |.((cn -FanMorphN ) . p).| = |.p.|

A10: (((q `1 ) / |.q.|) - cn) / (1 - cn) = (((q `1 ) / |.q.|) - cn) * ((1 - cn) " ) by XCMPLX_0:def 9
.= (((q `1 ) / |.q.|) - cn) * 0 by A9
.= 0 ;
then 1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ) = 1 ;
then (cn -FanMorphN ) . q = |[(|.q.| * 0 ),(|.q.| * 1)]| by A1, A10, Th56, SQUARE_1:83
.= |[0 ,|.q.|]| ;
then ( ((cn -FanMorphN ) . q) `2 = |.q.| & ((cn -FanMorphN ) . q) `1 = 0 ) by EUCLID:56;
then |.((cn -FanMorphN ) . p).| = sqrt ((|.q.| ^2 ) + (0 ^2 )) by JGRAPH_3:10
.= |.q.| by SQUARE_1:89 ;
hence |.((cn -FanMorphN ) . p).| = |.p.| ; :: thesis: verum

end;

suppose A11: 1 - cn <> 0 ; :: thesis: |.((cn -FanMorphN ) . p).| = |.p.|

per cases ( 1 - cn > 0 or 1 - cn < 0 ) by A11;

suppose A12: 1 - cn > 0 ; :: thesis: |.((cn -FanMorphN ) . p).| = |.p.|

- (1 - cn) <= - (((q `1 ) / |.q.|) - cn) by A8, XREAL_1:26;
then (- (1 - cn)) / (1 - cn) <= (- (((q `1 ) / |.q.|) - cn)) / (1 - cn) by A12, XREAL_1:74;
then - 1 <= (- (((q `1 ) / |.q.|) - cn)) / (1 - cn) by A12, XCMPLX_1:198;
then ((- (((q `1 ) / |.q.|) - cn)) / (1 - cn)) ^2 <= 1 ^2 by A5, A12, SQUARE_1:119;
then 1 - (((- (((q `1 ) / |.q.|) - cn)) / (1 - cn)) ^2 ) >= 0 by XREAL_1:50;
then A13: 1 - ((- ((((q `1 ) / |.q.|) - cn) / (1 - cn))) ^2 ) >= 0 by XCMPLX_1:188;
A14: (qz `2 ) ^2 = (|.q.| ^2 ) * ((sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 ))) ^2 ) by A3
.= (|.q.| ^2 ) * (1 - (((((q `1 ) / |.q.|) - cn) / (1 - cn)) ^2 )) by A13, SQUARE_1:def 4 ;
|.qz.| ^2 = ((qz `1 ) ^2 ) + ((qz `2 ) ^2 ) by JGRAPH_3:10
.= |.q.| ^2 by A4, A14 ;
then sqrt (|.qz.| ^2 ) = |.q.| by SQUARE_1:89;
hence |.((cn -FanMorphN ) . p).| = |.p.| by SQUARE_1:89; :: thesis: verum

end;

suppose A15: 1 - cn < 0 ; :: thesis: |.((cn -FanMorphN ) . p).| = |.p.|

0 + ((q `1 ) ^2 ) < ((q `1 ) ^2 ) + ((q `2 ) ^2 ) by A1, SQUARE_1:74, XREAL_1:10;
then ((q `1 ) ^2 ) / (|.q.| ^2 ) < (|.q.| ^2 ) / (|.q.| ^2 ) by A7, A6, XREAL_1:76;
then ((q `1 ) ^2 ) / (|.q.| ^2 ) < 1 by A7, XCMPLX_1:60;
then ((q `1 ) / |.q.|) ^2 < 1 by XCMPLX_1:77;
then A16: 1 > (q `1 ) / |.p.| by SQUARE_1:122;
((q `1 ) / |.q.|) - cn >= 0 by A1, XREAL_1:50;
hence |.((cn -FanMorphN ) . p).| = |.p.| by A15, A16, XREAL_1:11; :: thesis: verum

end;

end;

end;

end;

end;

suppose A17: ( (q `1 ) / |.q.| < cn & q `2 > 0 ) ; :: thesis: |.((cn -FanMorphN ) . p).| = |.p.|

then |.q.| <> 0 by JGRAPH_2:11, TOPRNS_1:25;
then A18: |.q.| ^2 > 0 by SQUARE_1:74;
A19: ((q `1 ) / |.q.|) - cn < 0 by A17, XREAL_1:51;
A20: |.q.| ^2 = ((q `1 ) ^2 ) + ((q `2 ) ^2 ) by JGRAPH_3:10;
0 <= (q `2 ) ^2 by XREAL_1:65;
then 0 + ((q `1 ) ^2 ) <= ((q `1 ) ^2 ) + ((q `2 ) ^2 ) by XREAL_1:9;
then ((q `1 ) ^2 ) / (|.q.| ^2 ) <= (|.q.| ^2 ) / (|.q.| ^2 ) by A20, XREAL_1:74;
then ((q `1 ) ^2 ) / (|.q.| ^2 ) <= 1 by A18, XCMPLX_1:60;
then ((q `1 ) / |.q.|) ^2 <= 1 by XCMPLX_1:77;
then - 1 <= (q `1 ) / |.q.| by SQUARE_1:121;
then A21: (- 1) - cn <= ((q `1 ) / |.q.|) - cn by XREAL_1:11;
A22: (cn -FanMorphN ) . q = |[(|.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))))]| by A17, Th57;
then A23: qz `2 = |.q.| * (sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))) by EUCLID:56;
A24: qz `1 = |.q.| * ((((q `1 ) / |.q.|) - cn) / (1 + cn)) by A22, EUCLID:56;

per cases ( 1 + cn = 0 or 1 + cn <> 0 ) ;

suppose A25: 1 + cn = 0 ; :: thesis: |.((cn -FanMorphN ) . p).| = |.p.|

(((q `1 ) / |.q.|) - cn) / (1 + cn) = (((q `1 ) / |.q.|) - cn) * ((1 + cn) " ) by XCMPLX_0:def 9
.= (((q `1 ) / |.q.|) - cn) * 0 by A25
.= 0 ;
then ( ((cn -FanMorphN ) . q) `2 = |.q.| & ((cn -FanMorphN ) . q) `1 = 0 ) by A22, EUCLID:56, SQUARE_1:83;
then |.((cn -FanMorphN ) . p).| = sqrt ((|.q.| ^2 ) + (0 ^2 )) by JGRAPH_3:10
.= |.q.| by SQUARE_1:89 ;
hence |.((cn -FanMorphN ) . p).| = |.p.| ; :: thesis: verum

end;

suppose A26: 1 + cn <> 0 ; :: thesis: |.((cn -FanMorphN ) . p).| = |.p.|

per cases ( 1 + cn > 0 or 1 + cn < 0 ) by A26;

suppose A27: 1 + cn > 0 ; :: thesis: |.((cn -FanMorphN ) . p).| = |.p.|

then (- (1 + cn)) / (1 + cn) <= (((q `1 ) / |.q.|) - cn) / (1 + cn) by A21, XREAL_1:74;
then - 1 <= (((q `1 ) / |.q.|) - cn) / (1 + cn) by A27, XCMPLX_1:198;
then ((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 <= 1 ^2 by A19, A27, SQUARE_1:119;
then A28: 1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ) >= 0 by XREAL_1:50;
A29: (qz `2 ) ^2 = (|.q.| ^2 ) * ((sqrt (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 ))) ^2 ) by A23
.= (|.q.| ^2 ) * (1 - (((((q `1 ) / |.q.|) - cn) / (1 + cn)) ^2 )) by A28, SQUARE_1:def 4 ;
|.qz.| ^2 = ((qz `1 ) ^2 ) + ((qz `2 ) ^2 ) by JGRAPH_3:10
.= |.q.| ^2 by A24, A29 ;
then sqrt (|.qz.| ^2 ) = |.q.| by SQUARE_1:89;
hence |.((cn -FanMorphN ) . p).| = |.p.| by SQUARE_1:89; :: thesis: verum

end;

suppose A30: 1 + cn < 0 ; :: thesis: |.((cn -FanMorphN ) . p).| = |.p.|

0 + ((q `1 ) ^2 ) < ((q `1 ) ^2 ) + ((q `2 ) ^2 ) by A17, SQUARE_1:74, XREAL_1:10;
then ((q `1 ) ^2 ) / (|.q.| ^2 ) < (|.q.| ^2 ) / (|.q.| ^2 ) by A18, A20, XREAL_1:76;
then ((q `1 ) ^2 ) / (|.q.| ^2 ) < 1 by A18, XCMPLX_1:60;
then ((q `1 ) / |.q.|) ^2 < 1 by XCMPLX_1:77;
then - 1 < (q `1 ) / |.p.| by SQUARE_1:122;
then A31: ((q `1 ) / |.q.|) - cn > (- 1) - cn by XREAL_1:11;
- (1 + cn) > - 0 by A30, XREAL_1:26;
hence |.((cn -FanMorphN ) . p).| = |.p.| by A17, A31, XREAL_1:51; :: thesis: verum

end;

end;

end;

end;

end;

suppose q `2 <= 0 ; :: thesis: |.((cn -FanMorphN ) . p).| = |.p.|

hence |.((cn -FanMorphN ) . p).| = |.p.| by Th56; :: thesis: verum

end;

end;