reserve x,y for set;
reserve C,C9,D,D9,E for non empty set;
reserve c for Element of C;
reserve c9 for Element of C9;
reserve d,d1,d2,d3,d4,e for Element of D;
reserve d9 for Element of D9;
reserve i,j for natural Number;
reserve F for Function of [:D,D9:],E;
reserve p,q for FinSequence of D,
  p9,q9 for FinSequence of D9;
reserve f,f9 for Function of C,D,
  h for Function of D,E;
reserve T,T1,T2,T3 for Tuple of i,D;
reserve T9 for Tuple of i, D9;
reserve S for Tuple of j, D;
reserve S9 for Tuple of j, D9;
reserve F,G for BinOp of D;
reserve u for UnOp of D;
reserve H for BinOp of E;

theorem Th75:
  F is associative & F is having_a_unity & e = the_unity_wrt F &
    F is having_an_inverseOp & G is_distributive_wrt F implies
      G[;](e,f) = C-->e
proof
  reconsider g = C-->e as Function of C,D;
  assume
A1: F is associative & F is having_a_unity & e = the_unity_wrt F & F is
  having_an_inverseOp & G is_distributive_wrt F;
  now
    let c;
    thus (G[;](e,f)).c = G.(e,f.c) by FUNCOP_1:53
      .= e by A1,Th66
      .= g.c;
  end;
  hence thesis by FUNCT_2:63;
end;
