Prove the following sequence of theorems. 

a. An automorphism of a field E carries elements that are squares of elements in E onto elements that are squares of elements of E. 

b. An automorphism of the field R of real numbers carries positive numbers onto positive numbers. 

c. If σ is an automorphism of R and a < b, where a, b ∈ R, then σ(a) < σ(b). 

d. The only automorphism of R is the identity automorphism.