let n be Element of NAT ; :: thesis: for x1, x2 being Element of REAL n
for L being Element of line_of_REAL n st x1 in L & x2 in L & x1 <> x2 holds
( Line x1,x2 = L & L is being_line )
let x1, x2 be Element of REAL n; :: thesis: for L being Element of line_of_REAL n st x1 in L & x2 in L & x1 <> x2 holds
( Line x1,x2 = L & L is being_line )
let L be Element of line_of_REAL n; :: thesis: ( x1 in L & x2 in L & x1 <> x2 implies ( Line x1,x2 = L & L is being_line ) )
assume that
A1: ( x1 in L & x2 in L ) and
A2: x1 <> x2 ; :: thesis: ( Line x1,x2 = L & L is being_line )
A3: Line x1,x2 c= L by A1, Th53;
L in line_of_REAL n ;
then ex L0 being Subset of (REAL n) st
( L = L0 & ex y1, y2 being Element of REAL n st L0 = Line y1,y2 ) ;
then L c= Line x1,x2 by A1, A2, EUCLID_4:12;
then Line x1,x2 = L by A3, XBOOLE_0:def 10;
hence ( Line x1,x2 = L & L is being_line ) by A2, EUCLID_4:def 2; :: thesis: verum