let V be RealLinearSpace; :: thesis: for u, v, x, y being VECTOR of V
for a being Real st Gen x,y holds
( pr1 (x,y,(u + v)) = (pr1 (x,y,u)) + (pr1 (x,y,v)) & pr2 (x,y,(u + v)) = (pr2 (x,y,u)) + (pr2 (x,y,v)) & pr1 (x,y,(u - v)) = (pr1 (x,y,u)) - (pr1 (x,y,v)) & pr2 (x,y,(u - v)) = (pr2 (x,y,u)) - (pr2 (x,y,v)) & pr1 (x,y,(a * u)) = a * (pr1 (x,y,u)) & pr2 (x,y,(a * u)) = a * (pr2 (x,y,u)) )

let u, v, x, y be VECTOR of V; :: thesis: for a being Real st Gen x,y holds
( pr1 (x,y,(u + v)) = (pr1 (x,y,u)) + (pr1 (x,y,v)) & pr2 (x,y,(u + v)) = (pr2 (x,y,u)) + (pr2 (x,y,v)) & pr1 (x,y,(u - v)) = (pr1 (x,y,u)) - (pr1 (x,y,v)) & pr2 (x,y,(u - v)) = (pr2 (x,y,u)) - (pr2 (x,y,v)) & pr1 (x,y,(a * u)) = a * (pr1 (x,y,u)) & pr2 (x,y,(a * u)) = a * (pr2 (x,y,u)) )

let a be Real; :: thesis: ( Gen x,y implies ( pr1 (x,y,(u + v)) = (pr1 (x,y,u)) + (pr1 (x,y,v)) & pr2 (x,y,(u + v)) = (pr2 (x,y,u)) + (pr2 (x,y,v)) & pr1 (x,y,(u - v)) = (pr1 (x,y,u)) - (pr1 (x,y,v)) & pr2 (x,y,(u - v)) = (pr2 (x,y,u)) - (pr2 (x,y,v)) & pr1 (x,y,(a * u)) = a * (pr1 (x,y,u)) & pr2 (x,y,(a * u)) = a * (pr2 (x,y,u)) ) )
assume A1: Gen x,y ; :: thesis: ( pr1 (x,y,(u + v)) = (pr1 (x,y,u)) + (pr1 (x,y,v)) & pr2 (x,y,(u + v)) = (pr2 (x,y,u)) + (pr2 (x,y,v)) & pr1 (x,y,(u - v)) = (pr1 (x,y,u)) - (pr1 (x,y,v)) & pr2 (x,y,(u - v)) = (pr2 (x,y,u)) - (pr2 (x,y,v)) & pr1 (x,y,(a * u)) = a * (pr1 (x,y,u)) & pr2 (x,y,(a * u)) = a * (pr2 (x,y,u)) )
set p1u = pr1 (x,y,u);
set p2u = pr2 (x,y,u);
set p1v = pr1 (x,y,v);
set p2v = pr2 (x,y,v);
A2: u = ((pr1 (x,y,u)) * x) + ((pr2 (x,y,u)) * y) by ;
A3: v = ((pr1 (x,y,v)) * x) + ((pr2 (x,y,v)) * y) by ;
then u + v = ((((pr1 (x,y,u)) * x) + ((pr2 (x,y,u)) * y)) + ((pr1 (x,y,v)) * x)) + ((pr2 (x,y,v)) * y) by
.= ((((pr1 (x,y,u)) * x) + ((pr1 (x,y,v)) * x)) + ((pr2 (x,y,u)) * y)) + ((pr2 (x,y,v)) * y) by RLVECT_1:def 3
.= (((pr1 (x,y,u)) * x) + ((pr1 (x,y,v)) * x)) + (((pr2 (x,y,u)) * y) + ((pr2 (x,y,v)) * y)) by RLVECT_1:def 3
.= (((pr1 (x,y,u)) + (pr1 (x,y,v))) * x) + (((pr2 (x,y,u)) * y) + ((pr2 (x,y,v)) * y)) by RLVECT_1:def 6
.= (((pr1 (x,y,u)) + (pr1 (x,y,v))) * x) + (((pr2 (x,y,u)) + (pr2 (x,y,v))) * y) by RLVECT_1:def 6 ;
hence ( pr1 (x,y,(u + v)) = (pr1 (x,y,u)) + (pr1 (x,y,v)) & pr2 (x,y,(u + v)) = (pr2 (x,y,u)) + (pr2 (x,y,v)) ) by ; :: thesis: ( pr1 (x,y,(u - v)) = (pr1 (x,y,u)) - (pr1 (x,y,v)) & pr2 (x,y,(u - v)) = (pr2 (x,y,u)) - (pr2 (x,y,v)) & pr1 (x,y,(a * u)) = a * (pr1 (x,y,u)) & pr2 (x,y,(a * u)) = a * (pr2 (x,y,u)) )
u - v = (((pr1 (x,y,u)) - (pr1 (x,y,v))) * x) + (((pr2 (x,y,u)) - (pr2 (x,y,v))) * y) by A2, A3, Lm1;
hence ( pr1 (x,y,(u - v)) = (pr1 (x,y,u)) - (pr1 (x,y,v)) & pr2 (x,y,(u - v)) = (pr2 (x,y,u)) - (pr2 (x,y,v)) ) by ; :: thesis: ( pr1 (x,y,(a * u)) = a * (pr1 (x,y,u)) & pr2 (x,y,(a * u)) = a * (pr2 (x,y,u)) )
a * u = ((a * (pr1 (x,y,u))) * x) + ((a * (pr2 (x,y,u))) * y) by ;
hence ( pr1 (x,y,(a * u)) = a * (pr1 (x,y,u)) & pr2 (x,y,(a * u)) = a * (pr2 (x,y,u)) ) by ; :: thesis: verum