let V be RealLinearSpace; :: thesis: ( ex u, v being VECTOR of V st
( ( for a, b being Real st (a * u) + (b * v) = 0. V holds
( a = 0 & b = 0 ) ) & ( for w being VECTOR of V ex a, b being Real st w = (a * u) + (b * v) ) ) implies OASpace V is OAffinPlane )
assume A1: ex u, v being VECTOR of V st
( ( for a, b being Real st (a * u) + (b * v) = 0. V holds
( a = 0 & b = 0 ) ) & ( for w being VECTOR of V ex a, b being Real st w = (a * u) + (b * v) ) ) ; :: thesis: OASpace V is OAffinPlane
set S = OASpace V;
consider u, v being VECTOR of V such that
A2: for a, b being Real st (a * u) + (b * v) = 0. V holds
( a = 0 & b = 0 ) and
A3: for w being VECTOR of V ex a, b being Real st w = (a * u) + (b * v) by A1;
for a, b, c, d being Element of (OASpace V) st not a,b // c,d & not a,b // d,c holds
ex p being Element of (OASpace V) st
( ( a,b // a,p or a,b // p,a ) & ( c,d // c,p or c,d // p,c ) ) by A3, Th36;
hence OASpace V is OAffinPlane by A2, Def6, Th38; :: thesis: verum