let n be Element of NAT ; :: thesis: for x being Element of REAL n
for L being Element of line_of_REAL n st not x in L & L is being_line holds
ex L0 being Element of line_of_REAL n st
( x in L0 & L0 _|_ L & L0 meets L )
let x be Element of REAL n; :: thesis: for L being Element of line_of_REAL n st not x in L & L is being_line holds
ex L0 being Element of line_of_REAL n st
( x in L0 & L0 _|_ L & L0 meets L )
let L be Element of line_of_REAL n; :: thesis: ( not x in L & L is being_line implies ex L0 being Element of line_of_REAL n st
( x in L0 & L0 _|_ L & L0 meets L ) )
assume ( not x in L & L is being_line ) ; :: thesis: ex L0 being Element of line_of_REAL n st
( x in L0 & L0 _|_ L & L0 meets L )
then consider x1, x2 being Element of REAL n such that
A1: ( L = Line x1,x2 & x - x1 _|_ x2 - x1 ) by Th59;
reconsider L0 = Line x1,x as Subset of (REAL n) ;
reconsider L0 = L0 as Element of line_of_REAL n by Th52;
A2: ( x in L0 & x1 in L0 & x1 in L ) by A1, EUCLID_4:10;
then A3: L0 meets L by Th54;
L0 _|_ L by A1, Def8;
hence ex L0 being Element of line_of_REAL n st
( x in L0 & L0 _|_ L & L0 meets L ) by A2, A3; :: thesis: verum