... ) ∧ ( 0 ≤ R β ≤ π / 4 ) ‒ First subcase: ( 0 < R α < π / 4 ) ∧ ( 0 < R β < π / 4 ) cos ⁡ ( R α + R β ) = cos ⁡ ( R α ) cos ⁡ ( R β ) − sin ⁡ ( R α ) sin ⁡ ( R β ) [Eq. (14.3...

... , it will hold that R c 2 = R a 2 + R b 2 , the last equation is known as Pythagoras’ theorem. Pythagoras’ theorem is a fundamental equation that appears throughout physics. Proof of Pythagoras’ theorem [Eq. ( 11.8 )]. Case: Fig. 11.7 − First subcase: value of ∡ x...

... + ( − 1 ) [Eq. (4.34)] , 1 + ( − R r ) ≠ 0 [Eq. (4.44)] , 1 − R r ≠ 0 [Eq. (4.45)] . − First subcase: n = 0, θ g e o m ( 1 , R r , n ) = θ g e o m ( 1 , R r , 0 ) (this subcase) , θ g...

... subcase: Z > 0 ⟹ R Z > 1 Z > 0 ⟹ ∃ n { ( Z = + n ) ∧ ( n ≠ 0 ) } , Z > 0 ⟹ ( Z = + n ) ∧ ( n ≠ 0 ) , R > 1 , Z > 0 ⟹ R > 1 , Z > 0 ⟹ 1 < R [Eq. (4.104)] , Z...

...). proof Case: ( G L n  is a non-vertical line ) ∧ ( R m  is the slope of G L n ) ∧ ( R b  is the y -intercept of G L n ) − First subcase: R_x ≠ 0 ( R x , R y ) ∈ G L n ⟺ R y...

... | < Z 2 ⟺ ( ⊖ Z 2 < Z 1 ) ∧ ( Z 1 < Z 2 ) . Proof: First case: | Z 1 | < Z 2 ⟹ ( ⊖ Z 2 < Z 1 ) ∧ ( Z 1 < Z 2 ) First subcase: Z 1 < ∅ (8.25) | Z 1 | < Z 2...

... 2 ) ⟺ Z 2 D 1 = Z 1 D 2 ; (9.1) ( Z 1 , D 1 ) ∼ ( Z 2 , D 2 ) ⟺ ( Z 2 , D 2 ) ∼ ( Z 1 , D 1 ) . Proof of transitivity: First case: Z₂ = 0 First subcase: ( Z 1 , D 1 ) ∼ ( Z 2...

... , Q B ∀ n n > N ⟹ α 1 ( n ) − α 2 ( n ) ≤ Q B < 0 } ⟹ [α₁(n)] < [α₂(n)] Subcase: ∃ N , Q B ∀ n n > N ⟹ α 1 ( n ) − α 2 ( n ) ≤ Q B < 0...

... 2 ⟹ R 3 ⊙ R 1 = R 3 ⊙ R 2 ) R 1 = R 2 ⟹ R 3 ⊙ R 1 = R 3 ⊙ R 2 ; Second case: ( R 3 ≠ ∅ ) ∧ ( R 3 ⊙ R 1 = R 3 ⊙ R 2 ⟹ R 1 = R 2 ) First subcase: (R₃ is positive...