reserve i,j,k,n for Nat;
reserve x,y,z for Tuple of n, BOOLEAN;

theorem Th1:
  for i,j,k holds j is prime & i<j & k<j & i<>0 implies ex a being
  Nat st (a*i) mod j = k & a < j
proof
  let i,j,k be Nat;
  assume that
A1: j is prime and
A2: i<j and
A3: k<j & i<>0;
  consider a being Nat such that
A4: (a*i) mod j = k by A1,A2,A3,Lm2;
  consider a9 being Nat such that
A5: a9 = a mod j;
  take a9;
  thus thesis by A2,A4,A5,EULER_2:8,NAT_D:1;
end;
