let V be RealLinearSpace; :: thesis: for w, y being VECTOR of V st Gen w,y holds
for u, v, v1 being VECTOR of V holds 2 * (PProJ w,y,u,(v # v1)) = (PProJ w,y,u,v) + (PProJ w,y,u,v1)
let w, y be VECTOR of V; :: thesis: ( Gen w,y implies for u, v, v1 being VECTOR of V holds 2 * (PProJ w,y,u,(v # v1)) = (PProJ w,y,u,v) + (PProJ w,y,u,v1) )
assume A1: Gen w,y ; :: thesis: for u, v, v1 being VECTOR of V holds 2 * (PProJ w,y,u,(v # v1)) = (PProJ w,y,u,v) + (PProJ w,y,u,v1)
let u, v, v1 be VECTOR of V; :: thesis: 2 * (PProJ w,y,u,(v # v1)) = (PProJ w,y,u,v) + (PProJ w,y,u,v1)
set a1 = pr1 w,y,u;
set a2 = pr2 w,y,u;
set b1 = pr1 w,y,v;
set b2 = pr2 w,y,v;
set c1 = pr1 w,y,v1;
set c2 = pr2 w,y,v1;
thus 2 * (PProJ w,y,u,(v # v1)) = ((pr1 w,y,u) * (2 * (pr1 w,y,(v # v1)))) + (2 * ((pr2 w,y,u) * (pr2 w,y,(v # v1))))
.= ((pr1 w,y,u) * ((pr1 w,y,v) + (pr1 w,y,v1))) + ((pr2 w,y,u) * (2 * (pr2 w,y,(v # v1)))) by A1, Lm18
.= ((pr1 w,y,u) * ((pr1 w,y,v) + (pr1 w,y,v1))) + ((pr2 w,y,u) * ((pr2 w,y,v) + (pr2 w,y,v1))) by A1, Lm18
.= (PProJ w,y,u,v) + (PProJ w,y,u,v1) ; :: thesis: verum