let n be Element of NAT ; :: thesis: for p1, p2, p3 being Point of (TOP-REAL n) holds plane p1,p2,p3 = { p where p is Point of (TOP-REAL n) : ex a1, a2, a3 being Real st
( (a1 + a2) + a3 = 1 & p = ((a1 * p1) + (a2 * p2)) + (a3 * p3) ) }
let p1, p2, p3 be Point of (TOP-REAL n); :: thesis: plane p1,p2,p3 = { p where p is Point of (TOP-REAL n) : ex a1, a2, a3 being Real st
( (a1 + a2) + a3 = 1 & p = ((a1 * p1) + (a2 * p2)) + (a3 * p3) ) }
thus plane p1,p2,p3 c= { p where p is Point of (TOP-REAL n) : ex a1, a2, a3 being Real st
( (a1 + a2) + a3 = 1 & p = ((a1 * p1) + (a2 * p2)) + (a3 * p3) ) } :: according to XBOOLE_0:def 10 :: thesis: { p where p is Point of (TOP-REAL n) : ex a1, a2, a3 being Real st
( (a1 + a2) + a3 = 1 & p = ((a1 * p1) + (a2 * p2)) + (a3 * p3) ) } c= plane p1,p2,p3

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in plane p1,p2,p3 or x in { p where p is Point of (TOP-REAL n) : ex a1, a2, a3 being Real st
( (a1 + a2) + a3 = 1 & p = ((a1 * p1) + (a2 * p2)) + (a3 * p3) ) } )
assume A1: x in plane p1,p2,p3 ; :: thesis: x in { p where p is Point of (TOP-REAL n) : ex a1, a2, a3 being Real st
( (a1 + a2) + a3 = 1 & p = ((a1 * p1) + (a2 * p2)) + (a3 * p3) ) }
then reconsider p0 = x as Point of (TOP-REAL n) ;
consider a1, a2, a3 being Real such that
A2: ( (a1 + a2) + a3 = 1 & p0 = ((a1 * p1) + (a2 * p2)) + (a3 * p3) ) by A1, Th57;
thus x in { p where p is Point of (TOP-REAL n) : ex a1, a2, a3 being Real st
( (a1 + a2) + a3 = 1 & p = ((a1 * p1) + (a2 * p2)) + (a3 * p3) ) } by A2; :: thesis: verum

end;

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in { p where p is Point of (TOP-REAL n) : ex a1, a2, a3 being Real st
( (a1 + a2) + a3 = 1 & p = ((a1 * p1) + (a2 * p2)) + (a3 * p3) ) } or x in plane p1,p2,p3 )
assume x in { p where p is Point of (TOP-REAL n) : ex a1, a2, a3 being Real st
( (a1 + a2) + a3 = 1 & p = ((a1 * p1) + (a2 * p2)) + (a3 * p3) ) } ; :: thesis: x in plane p1,p2,p3
then consider p being Point of (TOP-REAL n) such that
A3: ( p = x & ex a1, a2, a3 being Real st
( (a1 + a2) + a3 = 1 & p = ((a1 * p1) + (a2 * p2)) + (a3 * p3) ) ) ;
thus x in plane p1,p2,p3 by A3, Th61; :: thesis: verum