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