let V be RealLinearSpace; :: thesis: for x, y, u being VECTOR of V st Gen x,y holds
Ortm x,y,(- u) = - (Ortm x,y,u)
let x, y, u be VECTOR of V; :: thesis: ( Gen x,y implies Ortm x,y,(- u) = - (Ortm x,y,u) )
assume A1: Gen x,y ; :: thesis: Ortm x,y,(- u) = - (Ortm x,y,u)
then A2: - u = - (((pr1 x,y,u) * x) + ((pr2 x,y,u) * y)) by Lm5
.= (- ((pr1 x,y,u) * x)) + (- ((pr2 x,y,u) * y)) by RLVECT_1:45
.= ((pr1 x,y,u) * (- x)) + (- ((pr2 x,y,u) * y)) by RLVECT_1:39
.= ((- (pr1 x,y,u)) * x) + (- ((pr2 x,y,u) * y)) by RLVECT_1:38
.= ((- (pr1 x,y,u)) * x) + ((pr2 x,y,u) * (- y)) by RLVECT_1:39
.= ((- (pr1 x,y,u)) * x) + ((- (pr2 x,y,u)) * y) by RLVECT_1:38 ;
hence Ortm x,y,(- u) = ((- (pr1 x,y,u)) * x) + ((- (pr2 x,y,(- u))) * y) by A1, Lm6
.= ((- (pr1 x,y,u)) * x) + ((- (- (pr2 x,y,u))) * y) by A1, A2, Lm6
.= ((pr1 x,y,u) * (- x)) + ((- (- (pr2 x,y,u))) * y) by RLVECT_1:38
.= (- ((pr1 x,y,u) * x)) + ((- (- (pr2 x,y,u))) * y) by RLVECT_1:39
.= (- ((pr1 x,y,u) * x)) + ((- (pr2 x,y,u)) * (- y)) by RLVECT_1:38
.= (- ((pr1 x,y,u) * x)) + (- ((- (pr2 x,y,u)) * y)) by RLVECT_1:39
.= - (Ortm x,y,u) by RLVECT_1:45 ;
:: thesis: verum