
theorem Th9:
  for f,g being Function st ~[:f,g:] is one-to-one holds [:g,f:] is one-to-one
proof
  let f,g be Function such that
A1: ~[:f,g:] is one-to-one;
  let x1,x2 be object;
A2: dom[:g,f:] = [:dom g, dom f:] by FUNCT_3:def 8;
A3: dom[:f,g:] = [:dom f, dom g:] by FUNCT_3:def 8;
  assume x1 in dom[:g,f:];
  then consider x11,x12 being object such that
A4: x11 in dom g and
A5: x12 in dom f and
A6: x1 = [x11,x12] by A2,ZFMISC_1:84;
  assume x2 in dom[:g,f:];
  then consider x21,x22 being object such that
A7: x21 in dom g and
A8: x22 in dom f and
A9: x2 = [x21,x22] by A2,ZFMISC_1:84;
  x1 in dom[:g,f:] by A2,A4,A5,A6,ZFMISC_1:87;
  then
A10: x1 in dom~[:f,g:] by A2,A3,FUNCT_4:46;
  x2 in dom[:g,f:] by A2,A7,A8,A9,ZFMISC_1:87;
  then
A11: x2 in dom~[:f,g:] by A2,A3,FUNCT_4:46;
  assume
A12: [:g,f:].x1 = [:g,f:].x2;
A13: [:g,f:].(x11,x12) = [g.x11,f.x12] by A4,A5,FUNCT_3:def 8;
A14: [:g,f:].(x21,x22) = [g.x21,f.x22] by A7,A8,FUNCT_3:def 8;
  then
A15: f.x22 = f.x12 by A6,A9,A12,A13,XTUPLE_0:1;
A16: g.x11 = g.x21 by A6,A9,A12,A13,A14,XTUPLE_0:1;
  (~[:f,g:]).[x11,x12] = (~[:f,g:]).(x11,x12)
    .= [:f,g:].(x12,x11) by A6,A10,FUNCT_4:43
    .= [f.x22,g.x21] by A4,A5,A15,A16,FUNCT_3:def 8
    .= [:f,g:].(x22,x21) by A7,A8,FUNCT_3:def 8
    .= (~[:f,g:]).(x21,x22) by A9,A11,FUNCT_4:43
    .= (~[:f,g:]).[x21,x22];
  hence thesis by A1,A6,A9,A10,A11;
end;
