enrich elem with constants O : elem; I : elem; functions . + . : elem × elem → elem; . * . : elem × elem → elem prio 10 right; O- . : elem → elem; variables a : elem; b : elem; c : elem; axioms addcom: a + b = b + a; used for: s, ls; addzero-r: a + O = a; used for: s, ls; addinv-r: a + O- a = O; used for: s, ls; addassoc: (a + b) + c = a + b + c; used for: s, ls; multassoc: (a * b) * c = a * b * c; used for: s, ls; multone-r: a * I = a; used for: s, ls; multone-l: I * a = a; used for: s, ls; distr-r: a * (b + c) = a * b + a * c; distr-l: (b + c) * a = b * a + c * a; theorems multzero-r: a * O = O; used for: s, ls; multzero-l: O * a = O; used for: s, ls; addzero-l: O + a = a; used for: ls; addinv-l: O- a + a = O; used for: ls; trivialring: O = I → a = O; mult-minusone: O- I * a = O- a; used for: s, ls; end enrich