reserve a,b,m,x,y,i1,i2,i3,i for Integer,
  k,p,q,n for Nat,
  c,c1,c2 for Element of NAT,
  z for set;

theorem Th16:
  m>0 & a,m are_coprime implies ex n being Nat st (a*n-b) mod m = 0
proof
  assume that
A1: m>0 and
A2: a,m are_coprime;
  consider x being Integer such that
A3: (a*x-b) mod m = 0 by A2,Th15;
  consider q,n being Integer such that
A4: x=(m*q)+n and
A5: n>=0 and
  n<m by A1,Th13;
A6: (a*x-b) mod m = ((a*q)*m+(a*n-b)) mod m by A4
    .= (a*n-b) mod m by NAT_D:61;
  n in NAT by A5,INT_1:3;
  then reconsider n as Nat;
  take n;
  thus thesis by A3,A6;
end;
