let n be Nat; :: thesis: for a, r being Real
for y, z, x being Point of (TOP-REAL n)
for S, T, X being Element of REAL n st S = y & T = z & X = x & y <> z & y in Ball (x,r) & a = ((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S)))) holds
ex e being Point of (TOP-REAL n) st
( {e} = (halfline (y,z)) /\ (Sphere (x,r)) & e = ((1 - a) * y) + (a * z) )
let a, r be Real; :: thesis: for y, z, x being Point of (TOP-REAL n)
for S, T, X being Element of REAL n st S = y & T = z & X = x & y <> z & y in Ball (x,r) & a = ((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S)))) holds
ex e being Point of (TOP-REAL n) st
( {e} = (halfline (y,z)) /\ (Sphere (x,r)) & e = ((1 - a) * y) + (a * z) )
let y, z, x be Point of (TOP-REAL n); :: thesis: for S, T, X being Element of REAL n st S = y & T = z & X = x & y <> z & y in Ball (x,r) & a = ((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S)))) holds
ex e being Point of (TOP-REAL n) st
( {e} = (halfline (y,z)) /\ (Sphere (x,r)) & e = ((1 - a) * y) + (a * z) )
let S, T, X be Element of REAL n; :: thesis: ( S = y & T = z & X = x & y <> z & y in Ball (x,r) & a = ((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S)))) implies ex e being Point of (TOP-REAL n) st
( {e} = (halfline (y,z)) /\ (Sphere (x,r)) & e = ((1 - a) * y) + (a * z) ) )
assume that
A1: S = y and
A2: T = z and
A3: X = x ; :: thesis: ( not y <> z or not y in Ball (x,r) or not a = ((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S)))) or ex e being Point of (TOP-REAL n) st
( {e} = (halfline (y,z)) /\ (Sphere (x,r)) & e = ((1 - a) * y) + (a * z) ) )
set s = y;
set t = z;
A4: Sum (sqr (T - S)) >= 0 by RVSUM_1:86;
then A5: |.(T - S).| ^2 = Sum (sqr (T - S)) by SQUARE_1:def 2;
set A = Sum (sqr (T - S));
assume that
A6: y <> z and
A7: y in Ball (x,r) and
A8: a = ((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S)))) ; :: thesis: ex e being Point of (TOP-REAL n) st
( {e} = (halfline (y,z)) /\ (Sphere (x,r)) & e = ((1 - a) * y) + (a * z) )
A9: |.(T - S).| <> 0 by A1, A2, A6, EUCLID:16;

A10: now :: thesis: not Sum (sqr (T - S)) <= 0

assume Sum (sqr (T - S)) <= 0 ; :: thesis: contradiction
then Sum (sqr (T - S)) = 0 by RVSUM_1:86;
hence contradiction by A9; :: thesis: verum

end;

set C = (Sum (sqr (S - X))) - (r ^2);
set B = 2 * |((z - y),(y - x))|;
A11: ( r = 0 or r <> 0 ) ;
Sum (sqr (S - X)) >= 0 by RVSUM_1:86;
then A12: |.(S - X).| ^2 = Sum (sqr (S - X)) by SQUARE_1:def 2;
|.(y - x).| < r by A7, Th5;
then (sqrt (Sum (sqr (S - X)))) ^2 < r ^2 by A1, A3, SQUARE_1:16;
then A13: (Sum (sqr (S - X))) - (r ^2) < 0 by A12, XREAL_1:49;
then A14: ((Sum (sqr (S - X))) - (r ^2)) / (Sum (sqr (T - S))) < 0 by A10, XREAL_1:141;
set l2 = ((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))));
set l1 = ((- (2 * |((z - y),(y - x))|)) - (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))));
take e1 = ((1 - (((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S)))))) * y) + ((((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))) * z); :: thesis: ( {e1} = (halfline (y,z)) /\ (Sphere (x,r)) & e1 = ((1 - a) * y) + (a * z) )
A15: 0 <= - (- r) by A7;
A16: ( delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2))) = ((2 * |((z - y),(y - x))|) ^2) - ((4 * (Sum (sqr (T - S)))) * ((Sum (sqr (S - X))) - (r ^2))) & (2 * |((z - y),(y - x))|) ^2 >= 0 ) by QUIN_1:def 1, XREAL_1:63;
A17: for x being Real holds Polynom ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)),x) = Quard ((Sum (sqr (T - S))),(((- (2 * |((z - y),(y - x))|)) - (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))),(((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))),x)

proof

let x be Real; :: thesis: Polynom ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)),x) = Quard ((Sum (sqr (T - S))),(((- (2 * |((z - y),(y - x))|)) - (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))),(((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))),x)
thus Polynom ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)),x) = (((Sum (sqr (T - S))) * (x ^2)) + ((2 * |((z - y),(y - x))|) * x)) + ((Sum (sqr (S - X))) - (r ^2)) by POLYEQ_1:def 2
.= ((Sum (sqr (T - S))) * (x - (((- (2 * |((z - y),(y - x))|)) - (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))))) * (x - (((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S)))))) by A10, A13, A16, QUIN_1:16
.= (Sum (sqr (T - S))) * ((x - (((- (2 * |((z - y),(y - x))|)) - (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S)))))) * (x - (((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S)))))))
.= Quard ((Sum (sqr (T - S))),(((- (2 * |((z - y),(y - x))|)) - (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))),(((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))),x) by POLYEQ_1:def 3 ; :: thesis: verum

end;

then ((Sum (sqr (S - X))) - (r ^2)) / (Sum (sqr (T - S))) = (((- (2 * |((z - y),(y - x))|)) - (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))) * (((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))) by A10, POLYEQ_1:11;
then A18: ( ( ((- (2 * |((z - y),(y - x))|)) - (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S)))) < 0 & ((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S)))) > 0 ) or ( ((- (2 * |((z - y),(y - x))|)) - (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S)))) > 0 & ((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S)))) < 0 ) ) by A14, XREAL_1:133;
A19: (((Sum (sqr (T - S))) * ((((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))) ^2)) + ((2 * |((z - y),(y - x))|) * (((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))))) - (- ((Sum (sqr (S - X))) - (r ^2))) = Polynom ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)),(((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S)))))) by POLYEQ_1:def 2
.= Quard ((Sum (sqr (T - S))),(((- (2 * |((z - y),(y - x))|)) - (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))),(((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))),(((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S)))))) by A17
.= (Sum (sqr (T - S))) * (((((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))) - (((- (2 * |((z - y),(y - x))|)) - (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S)))))) * ((((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))) - (((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))))) by POLYEQ_1:def 3
.= 0 ;
|.(e1 - x).| ^2 = |.((((1 * y) - ((((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))) * y)) + ((((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))) * z)) - x).| ^2 by RLVECT_1:35
.= |.(((y - ((((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))) * y)) + ((((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))) * z)) - x).| ^2 by RLVECT_1:def 8
.= |.(((y + ((((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))) * z)) - ((((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))) * y)) - x).| ^2 by RLVECT_1:def 3
.= |.((y + (((((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))) * z) - ((((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))) * y))) - x).| ^2 by RLVECT_1:def 3
.= |.((y - x) + (((((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))) * z) - ((((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))) * y))).| ^2 by RLVECT_1:def 3
.= |.((y - x) + ((((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))) * (z - y))).| ^2 by RLVECT_1:34
.= ((|.(y - x).| ^2) + (2 * |(((((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))) * (z - y)),(y - x))|)) + (|.((((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))) * (z - y)).| ^2) by EUCLID_2:45
.= ((Sum (sqr (S - X))) + (2 * ((((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))) * |((z - y),(y - x))|))) + (|.((((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))) * (z - y)).| ^2) by A12, A1, A3, EUCLID_2:19
.= ((Sum (sqr (S - X))) + ((2 * (((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S)))))) * |((z - y),(y - x))|)) + ((|.(((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))).| * |.(z - y).|) ^2) by TOPRNS_1:7
.= ((Sum (sqr (S - X))) + ((2 * (((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S)))))) * |((z - y),(y - x))|)) + ((|.(((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))).| ^2) * (|.(z - y).| ^2))
.= ((Sum (sqr (S - X))) + ((((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))) * (2 * |((z - y),(y - x))|))) + (((((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))) ^2) * (|.(T - S).| ^2)) by A1, A2, COMPLEX1:75
.= ((Sum (sqr (S - X))) + ((((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))) * (2 * |((z - y),(y - x))|))) + (((((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S))))) ^2) * (Sum (sqr (T - S)))) by A4, SQUARE_1:def 2
.= r ^2 by A19 ;
then ( |.(e1 - x).| = r or |.(e1 - x).| = - r ) by SQUARE_1:40;
then A20: e1 in Sphere (x,r) by A15, A11;
A21: (4 * (Sum (sqr (T - S)))) * ((Sum (sqr (S - X))) - (r ^2)) < 0 by A10, A13, XREAL_1:129, XREAL_1:132;
then A22: e1 in halfline (y,z) by A10, A16, A18, QUIN_1:25;

hereby :: according to TARSKI:def 3,XBOOLE_0:def 10 :: thesis: ( (halfline (y,z)) /\ (Sphere (x,r)) c= {e1} & e1 = ((1 - a) * y) + (a * z) )

let d be object ; :: thesis: ( d in {e1} implies d in (halfline (y,z)) /\ (Sphere (x,r)) )
assume d in {e1} ; :: thesis: d in (halfline (y,z)) /\ (Sphere (x,r))
then d = e1 by TARSKI:def 1;
hence d in (halfline (y,z)) /\ (Sphere (x,r)) by A22, A20, XBOOLE_0:def 4; :: thesis: verum

end;

hereby :: according to TARSKI:def 3 :: thesis: e1 = ((1 - a) * y) + (a * z)

let d be object ; :: thesis: ( d in (halfline (y,z)) /\ (Sphere (x,r)) implies d in {e1} )
assume A23: d in (halfline (y,z)) /\ (Sphere (x,r)) ; :: thesis: d in {e1}
then d in halfline (y,z) by XBOOLE_0:def 4;
then consider l being Real such that
A24: d = ((1 - l) * y) + (l * z) and
A25: 0 <= l ;
A26: |.(l * (z - y)).| ^2 = (|.l.| * |.(z - y).|) ^2 by TOPRNS_1:7
.= (|.l.| ^2) * (|.(z - y).| ^2)
.= (l ^2) * (|.(z - y).| ^2) by COMPLEX1:75 ;
d in Sphere (x,r) by A23, XBOOLE_0:def 4;
then r = |.((((1 - l) * y) + (l * z)) - x).| by A24, Th7
.= |.((((1 * y) - (l * y)) + (l * z)) - x).| by RLVECT_1:35
.= |.(((y - (l * y)) + (l * z)) - x).| by RLVECT_1:def 8
.= |.((y - ((l * y) - (l * z))) - x).| by RLVECT_1:29
.= |.((y + (- ((l * y) - (l * z)))) + (- x)).|
.= |.((y + (- x)) + (- ((l * y) - (l * z)))).| by RLVECT_1:def 3
.= |.((y - x) - ((l * y) - (l * z))).|
.= |.((y + (- x)) + (- ((l * y) - (l * z)))).|
.= |.((y - x) + (- (l * (y - z)))).| by RLVECT_1:34
.= |.((y - x) + (l * (- (y - z)))).| by RLVECT_1:25
.= |.((y - x) + (l * (z - y))).| by RLVECT_1:33 ;
then r ^2 = ((|.(y - x).| ^2) + (2 * |((l * (z - y)),(y - x))|)) + (|.(l * (z - y)).| ^2) by EUCLID_2:45
.= ((|.(y - x).| ^2) + (2 * (l * |((z - y),(y - x))|))) + (|.(l * (z - y)).| ^2) by EUCLID_2:19 ;
then (((Sum (sqr (T - S))) * (l ^2)) + ((2 * |((z - y),(y - x))|) * l)) + ((Sum (sqr (S - X))) - (r ^2)) = 0 by A5, A12, A1, A3, A2, A26;
then Polynom ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)),l) = 0 by POLYEQ_1:def 2;
then ( l = ((- (2 * |((z - y),(y - x))|)) - (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S)))) or l = ((- (2 * |((z - y),(y - x))|)) + (sqrt (delta ((Sum (sqr (T - S))),(2 * |((z - y),(y - x))|),((Sum (sqr (S - X))) - (r ^2)))))) / (2 * (Sum (sqr (T - S)))) ) by A10, A13, A16, POLYEQ_1:5;
hence d in {e1} by A10, A21, A16, A18, A24, A25, QUIN_1:25, TARSKI:def 1; :: thesis: verum

end;

thus e1 = ((1 - a) * y) + (a * z) by A8; :: thesis: verum