let AS be AffinSpace; :: thesis: for a being Element of AS

for A being Subset of AS st A is being_line & a in A holds

a * A = A

let a be Element of AS; :: thesis: for A being Subset of AS st A is being_line & a in A holds

a * A = A

let A be Subset of AS; :: thesis: ( A is being_line & a in A implies a * A = A )

assume that

A1: A is being_line and

A2: a in A ; :: thesis: a * A = A

A // A by A1, AFF_1:41;

hence a * A = A by A1, A2, Def3; :: thesis: verum

for A being Subset of AS st A is being_line & a in A holds

a * A = A

let a be Element of AS; :: thesis: for A being Subset of AS st A is being_line & a in A holds

a * A = A

let A be Subset of AS; :: thesis: ( A is being_line & a in A implies a * A = A )

assume that

A1: A is being_line and

A2: a in A ; :: thesis: a * A = A

A // A by A1, AFF_1:41;

hence a * A = A by A1, A2, Def3; :: thesis: verum