let n be Element of NAT ; :: thesis: for x being Element of REAL n
for L being Element of line_of_REAL n
for P0, P1 being Element of plane_of_REAL n st not x in L & L is being_line & x in P0 & L c= P0 & x in P1 & L c= P1 holds
P0 = P1
let x be Element of REAL n; :: thesis: for L being Element of line_of_REAL n
for P0, P1 being Element of plane_of_REAL n st not x in L & L is being_line & x in P0 & L c= P0 & x in P1 & L c= P1 holds
P0 = P1
let L be Element of line_of_REAL n; :: thesis: for P0, P1 being Element of plane_of_REAL n st not x in L & L is being_line & x in P0 & L c= P0 & x in P1 & L c= P1 holds
P0 = P1
let P0, P1 be Element of plane_of_REAL n; :: thesis: ( not x in L & L is being_line & x in P0 & L c= P0 & x in P1 & L c= P1 implies P0 = P1 )
assume A1: ( not x in L & L is being_line & x in P0 & L c= P0 & x in P1 & L c= P1 ) ; :: thesis: P0 = P1
then consider x1, x2 being Element of REAL n such that
A2: ( L = Line x1,x2 & x - x1,x2 - x1 are_lindependent2 ) by Th60;
A3: ( x1 in L & x2 in L ) by A2, EUCLID_4:10;
then P0 = plane x1,x,x2 by A1, A2, Th97;
hence P0 = P1 by A1, A2, A3, Th97; :: thesis: verum