defpred S₁[ Real, set ] means $2 = |[(cos (((2 * PI) * r) * $1)),(sin (((2 * PI) * r) * $1))]|;
A1: for x being Element of I[01] ex y being Element of the carrier of (Tunit_circle 2) st S₁[x,y]

proof

let x be Element of I[01]; :: thesis: ex y being Element of the carrier of (Tunit_circle 2) st S₁[x,y]
set y = |[(cos (((2 * PI) * r) * x)),(sin (((2 * PI) * r) * x))]|;
|.(|[(cos (((2 * PI) * r) * x)),(sin (((2 * PI) * r) * x))]| - |[0,0]|).| = |.|[(cos (((2 * PI) * r) * x)),(sin (((2 * PI) * r) * x))]|.| by EUCLID:54, RLVECT_1:13
.= sqrt (((|[(cos (((2 * PI) * r) * x)),(sin (((2 * PI) * r) * x))]| `1) ^2) + ((|[(cos (((2 * PI) * r) * x)),(sin (((2 * PI) * r) * x))]| `2) ^2)) by JGRAPH_1:30
.= sqrt (((cos (((2 * PI) * r) * x)) ^2) + ((|[(cos (((2 * PI) * r) * x)),(sin (((2 * PI) * r) * x))]| `2) ^2)) by EUCLID:52
.= sqrt (((cos (((2 * PI) * r) * x)) ^2) + ((sin (((2 * PI) * r) * x)) ^2)) by EUCLID:52
.= 1 by SIN_COS:29, SQUARE_1:18 ;
then reconsider y = |[(cos (((2 * PI) * r) * x)),(sin (((2 * PI) * r) * x))]| as Element of the carrier of (Tunit_circle 2) by Lm11, TOPREAL9:9;
take y ; :: thesis: S₁[x,y]
thus S₁[x,y] ; :: thesis: verum

end;

ex f being Function of the carrier of I[01], the carrier of (Tunit_circle 2) st
for x being Element of I[01] holds S₁[x,f . x] from FUNCT_2:sch 3(A1);
hence ex b₁ being Function of I[01],(Tunit_circle 2) st
for x being Point of I[01] holds b₁ . x = |[(cos (((2 * PI) * r) * x)),(sin (((2 * PI) * r) * x))]| ; :: thesis: verum