reserve x1, x2, x3, x4, x5, x6, x7 for set;

theorem
  x1, x2, x3 are_mutually_distinct & x4, x5, x6 are_mutually_distinct
  & { x1, x2, x3 } misses { x4, x5, x6 } implies x1, x2, x3, x4, x5, x6
  are_mutually_distinct
proof
  assume that
A1: x1, x2, x3 are_mutually_distinct and
A2: x4, x5, x6 are_mutually_distinct and
A3: { x1, x2, x3 } misses { x4, x5, x6 };
A4: x1 <> x2 by A1,ZFMISC_1:def 5;
A5: x3 <> x5 by A3,Th4;
A6: x3 <> x4 by A3,Th4;
A7: x1 <> x6 by A3,Th4;
A8: x1 <> x3 by A1,ZFMISC_1:def 5;
A9: x2 <> x5 by A3,Th4;
A10: x5 <> x6 by A2,ZFMISC_1:def 5;
A11: x2 <> x4 by A3,Th4;
A12: x4 <> x5 by A2,ZFMISC_1:def 5;
A13: x2 <> x6 by A3,Th4;
A14: x4 <> x6 by A2,ZFMISC_1:def 5;
A15: x3 <> x6 by A3,Th4;
A16: x1 <> x5 by A3,Th4;
A17: x2 <> x3 by A1,ZFMISC_1:def 5;
  x1 <> x4 by A3,Th4;
  hence thesis by A4,A17,A8,A12,A10,A14,A16,A7,A11,A9,A13,A6,A5,A15,
ZFMISC_1:def 8;
end;
