Sentence examples for and by commutativity of from inspiring English sources

Since and, by commutativity of and, we have (261).

On the other hand, since gx = F x, y) and gy = F y, x), by commutativity of F and g, we have g ( g x ) = g ( F ( x, y ) ) = F ( g x, g y ), g ( g y ) = g ( F ( y, x ) ) = F ( g y, g x ).

Since g x = F ( x, y ) and g y = F ( y, x ), by commutativity of F and g, we have g ( g x ) = g ( F ( x, y ) ) = F ( g x, g y ) and g ( g y ) = g ( F ( y, x ) ) = F ( g y, g x ).

Since g x = F ( x, y, z, w ), g y = F ( y, z, w, x ), g z = F ( z, w, x, y ), and g z = F ( z, w, x, y ), by commutativity of F and g, we have.

By commutativity of A and by equation (1), we have y ∈ W ( x ) ⊆ y ∈ W ( T ( x ) ) = T ( W ( x ) ) ⊆ T ( y ), which is a contradiction.

Since G X = F X, by commutativity of f and g, we have G G X = G F X = F G X .Let G X = A. Then G A = F A. Thus, A is another coincidence point of f and g.

By commutativity of Diagram (11), we have (pi ^mathrm{un}((h x,n_i)) _x)=n_i).

Again, by commutativity of (11), we have ((w x,n_i))_v in ker (pi ^mathrm{un})) for all (v in X^*).

On the other hand, (pi (k) = 1) implies ( pi (k_v) = 1) for all (v in X^n); moreover, the (S_d^{(n)} -coordinate of (varphi _n(k)) is (1), by commutativity of Diagram (14).

By commutativity of diagram (8), the map (rsGamma:Xrightrightarrows X) also has a fixed point (x_{omega}=rs y_{omega})), (y_{omega}in Gamma (x_{omega})) satisfying ({x_{omega},y_{omega}}) are ω-near, i.e., (x_{omega}) is an ω-fixed point for Γ.

Since g x=F x,y) and g y=F y,x), by the commutativity of F and g, we have g ( g ( x ) ) = g ( F ( x, y ) ) = F ( gx, gy ), and g ( gy ) = g ( F ( y, x ) ) = F ( gy, gx ).