let AS be AffinSpace; :: thesis: for a, b being Element of AS
for A, C being Subset of AS st a,b // A & A // C holds
a,b // C
let a, b be Element of AS; :: thesis: for A, C being Subset of AS st a,b // A & A // C holds
a,b // C
let A, C be Subset of AS; :: thesis: ( a,b // A & A // C implies a,b // C )
assume that
A1: a,b // A and
A2: A // C ; :: thesis: a,b // C
consider p, q, c, d being Element of AS such that
A3: p <> q and
A4: c <> d and
A5: p,q // c,d and
A6: A = Line (p,q) and
A7: C = Line (c,d) by A2, Th36;
A8: q in A by A6, Th14;
A9: A is being_line by A2;
p in A by A6, Th14;
then a,b // p,q by A1, A3, A8, A9, Th26;
then a,b // c,d by A3, A5, Th4;
hence a,b // C by A4, A7; :: thesis: verum