let V be RealLinearSpace; :: thesis: for x, y, u, v being VECTOR of V st Gen x,y holds
Orte x,y,(u + v) = (Orte x,y,u) + (Orte x,y,v)
let x, y, u, v be VECTOR of V; :: thesis: ( Gen x,y implies Orte x,y,(u + v) = (Orte x,y,u) + (Orte x,y,v) )
assume A1: Gen x,y ; :: thesis: Orte x,y,(u + v) = (Orte x,y,u) + (Orte x,y,v)
hence Orte x,y,(u + v) = ((pr2 x,y,(u + v)) * x) + ((- ((pr1 x,y,u) + (pr1 x,y,v))) * y) by Lm7
.= (((pr2 x,y,u) + (pr2 x,y,v)) * x) + ((- ((pr1 x,y,u) + (pr1 x,y,v))) * y) by A1, Lm7
.= (((pr2 x,y,u) * x) + ((pr2 x,y,v) * x)) + ((- ((pr1 x,y,u) + (pr1 x,y,v))) * y) by RLVECT_1:def 9
.= (((pr2 x,y,u) * x) + ((pr2 x,y,v) * x)) + (((pr1 x,y,u) + (pr1 x,y,v)) * (- y)) by RLVECT_1:38
.= (((pr2 x,y,u) * x) + ((pr2 x,y,v) * x)) + (- (((pr1 x,y,u) + (pr1 x,y,v)) * y)) by RLVECT_1:39
.= (((pr2 x,y,u) * x) + ((pr2 x,y,v) * x)) + (- (((pr1 x,y,u) * y) + ((pr1 x,y,v) * y))) by RLVECT_1:def 9
.= (((pr2 x,y,u) * x) + ((pr2 x,y,v) * x)) + ((- ((pr1 x,y,u) * y)) + (- ((pr1 x,y,v) * y))) by RLVECT_1:45
.= ((pr2 x,y,u) * x) + (((pr2 x,y,v) * x) + ((- ((pr1 x,y,u) * y)) + (- ((pr1 x,y,v) * y)))) by RLVECT_1:def 6
.= ((pr2 x,y,u) * x) + ((- ((pr1 x,y,u) * y)) + (((pr2 x,y,v) * x) + (- ((pr1 x,y,v) * y)))) by RLVECT_1:def 6
.= (((pr2 x,y,u) * x) + (- ((pr1 x,y,u) * y))) + (((pr2 x,y,v) * x) + (- ((pr1 x,y,v) * y))) by RLVECT_1:def 6
.= (((pr2 x,y,u) * x) + ((pr1 x,y,u) * (- y))) + (((pr2 x,y,v) * x) + (- ((pr1 x,y,v) * y))) by RLVECT_1:39
.= (((pr2 x,y,u) * x) + ((pr1 x,y,u) * (- y))) + (((pr2 x,y,v) * x) + ((pr1 x,y,v) * (- y))) by RLVECT_1:39
.= (((pr2 x,y,u) * x) + ((- (pr1 x,y,u)) * y)) + (((pr2 x,y,v) * x) + ((pr1 x,y,v) * (- y))) by RLVECT_1:38
.= (Orte x,y,u) + (Orte x,y,v) by RLVECT_1:38 ;
:: thesis: verum