let n be Nat; :: thesis: for x being Element of REAL n
for L being Element of line_of_REAL n
for P being Element of plane_of_REAL n st x in P & L c= P & not x in L & L is being_line holds
P is being_plane
let x be Element of REAL n; :: thesis: for L being Element of line_of_REAL n
for P being Element of plane_of_REAL n st x in P & L c= P & not x in L & L is being_line holds
P is being_plane
let L be Element of line_of_REAL n; :: thesis: for P being Element of plane_of_REAL n st x in P & L c= P & not x in L & L is being_line holds
P is being_plane
let P be Element of plane_of_REAL n; :: thesis: ( x in P & L c= P & not x in L & L is being_line implies P is being_plane )
assume A1: ( x in P & L c= P ) ; :: thesis: ( x in L or not L is being_line or P is being_plane )
assume ( not x in L & L is being_line ) ; :: thesis: P is being_plane
then consider x1, x2 being Element of REAL n such that
A2: L = Line (x1,x2) and
A3: x - x1,x2 - x1 are_lindependent2 by Th55;
( x1 in L & x2 in L ) by A2, EUCLID_4:9;
then P = plane (x1,x,x2) by A1, A3, Th92;
hence P is being_plane by A3; :: thesis: verum