defpred S₁[ object , object ] means for p being Point of (TOP-REAL 2) st p = $1 holds
$2 = tricord1 (p1,p2,p3,p);
set X = the carrier of (TOP-REAL 2);
set Y = the carrier of R^1;
A1: for x being object st x in the carrier of (TOP-REAL 2) holds
ex y being object st
( y in the carrier of R^1 & S₁[x,y] )

proof

let x be object ; :: thesis: ( x in the carrier of (TOP-REAL 2) implies ex y being object st
( y in the carrier of R^1 & S₁[x,y] ) )
assume x in the carrier of (TOP-REAL 2) ; :: thesis: ex y being object st
( y in the carrier of R^1 & S₁[x,y] )
then reconsider p0 = x as Point of (TOP-REAL 2) ;
A2: tricord1 (p1,p2,p3,p0) in REAL by XREAL_0:def 1;
S₁[x, tricord1 (p1,p2,p3,p0)] ;
hence ex y being object st
( y in the carrier of R^1 & S₁[x,y] ) by A2, TOPMETR:17; :: thesis: verum

end;

ex f being Function of the carrier of (TOP-REAL 2), the carrier of R^1 st
for x being object st x in the carrier of (TOP-REAL 2) holds
S₁[x,f . x] from FUNCT_2:sch 1(A1);
then consider g being Function of the carrier of (TOP-REAL 2), the carrier of R^1 such that
A3: for x being object st x in the carrier of (TOP-REAL 2) holds
for p being Point of (TOP-REAL 2) st p = x holds
g . x = tricord1 (p1,p2,p3,p) ;
reconsider f0 = g as Function of (TOP-REAL 2),R^1 ;
for p being Point of (TOP-REAL 2) holds f0 . p = tricord1 (p1,p2,p3,p) by A3;
hence ex b₁ being Function of (TOP-REAL 2),R^1 st
for p being Point of (TOP-REAL 2) holds b₁ . p = tricord1 (p1,p2,p3,p) ; :: thesis: verum