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;
reserve fp,fp1 for FinSequence of NAT,

  b,c,d, n for Element of NAT,
  a for Nat;
reserve i,m,m1,m2,m3,r,s,a,b,c,c1,c2,x,y for Integer;

theorem Th41:
  m1 <> 0 & m2 <> 0 & not (m1 gcd m2) divides (c1-c2) implies not
  ex x st (x - c1) mod m1 = 0 & (x - c2) mod m2 = 0
proof
  assume that
A1: m1 <> 0 and
A2: m2 <> 0 and
A3: not (m1 gcd m2) divides (c1-c2);
A4: (m1 gcd m2) divides m2 by INT_2:21;
  given x such that
A5: (x - c1) mod m1 = 0 and
A6: (x - c2) mod m2 = 0;
  m2 divides (x-c2) by A2,A6,Lm10;
  then
A7: (m1 gcd m2) divides (x-c2) by A4,INT_2:9;
A8: (m1 gcd m2) divides m1 by INT_2:21;
  m1 divides (x-c1) by A1,A5,Lm10;
  then (m1 gcd m2) divides (x-c1) by A8,INT_2:9;
  then (m1 gcd m2) divides ((x-c2)-(x-c1)) by A7,Lm4;
  hence contradiction by A3;
end;
