reserve a,b,i,k,m,n for Nat;
reserve s,z for non zero Nat;
reserve c for Complex;

theorem
  (ex k st a = 7*k+1 or a = 7*k+2 or a = 7*k+4) &
  (ex k st b = 7*k+1 or b = 7*k+2 or b = 7*k+4) implies
  ex k st a+b = 7*k+1 or ... or a+b = 7*k+6
  proof
    given k1 being Nat such that
A1: a = 7*k1+1 or a = 7*k1+2 or a = 7*k1+4;
    given k2 being Nat such that
A2: b = 7*k2+1 or b = 7*k2+2 or b = 7*k2+4;
    per cases by A1,A2;
    suppose
A3:   a = 7*k1+1 & b = 7*k2+1;
      ex k st a+b = 7*k+(1 qua Nat) or ... or a+b = 7*k+(6 qua Nat)
      proof
        take k1+k2;
        thus thesis by A3;
      end;
      hence thesis;
    end;
    suppose
A4:   a = 7*k1+1 & b = 7*k2+2;
      ex k st a+b = 7*k+(1 qua Nat) or ... or a+b = 7*k+(6 qua Nat)
      proof
        take k1+k2;
        thus thesis by A4;
      end;
      hence thesis;
    end;
    suppose
A5:   a = 7*k1+1 & b = 7*k2+4;
      ex k st a+b = 7*k+(1 qua Nat) or ... or a+b = 7*k+(6 qua Nat)
      proof
        take k1+k2;
        thus thesis by A5;
      end;
      hence thesis;
    end;
    suppose
A6:   a = 7*k1+2 & b = 7*k2+1;
      ex k st a+b = 7*k+(1 qua Nat) or ... or a+b = 7*k+(6 qua Nat)
      proof
        take k1+k2;
        thus thesis by A6;
      end;
      hence thesis;
    end;
    suppose
A7:   a = 7*k1+2 & b = 7*k2+2;
      ex k st a+b = 7*k+(1 qua Nat) or ... or a+b = 7*k+(6 qua Nat)
      proof
        take k1+k2;
        thus thesis by A7;
      end;
      hence thesis;
    end;
    suppose
A8:   a = 7*k1+2 & b = 7*k2+4;
      ex k st a+b = 7*k+(1 qua Nat) or ... or a+b = 7*k+(6 qua Nat)
      proof
        take k1+k2;
        thus thesis by A8;
      end;
      hence thesis;
    end;
    suppose
A9:   a = 7*k1+4 & b = 7*k2+1;
      ex k st a+b = 7*k+(1 qua Nat) or ... or a+b = 7*k+(6 qua Nat)
      proof
        take k1+k2;
        thus thesis by A9;
      end;
      hence thesis;
    end;
    suppose
A10:   a = 7*k1+4 & b = 7*k2+2;
      ex k st a+b = 7*k+(1 qua Nat) or ... or a+b = 7*k+(6 qua Nat)
      proof
        take k1+k2;
        thus thesis by A10;
      end;
      hence thesis;
    end;
    suppose
A11:   a = 7*k1+4 & b = 7*k2+4;
      ex k st a+b = 7*k+(1 qua Nat) or ... or a+b = 7*k+(6 qua Nat)
      proof
        take k1+k2+1;
        thus thesis by A11;
      end;
      hence thesis;
    end;
  end;
