
theorem Th17:
for r be Real, X be set, g be PartFunc of X,REAL holds
 less_dom(g,r) = g"(].-infty,r.[)
proof
    let r be Real, X be set, g be PartFunc of X,REAL;
    for z be object holds
     z in less_dom(g,r) iff z in g"(].-infty,r.[)
    proof
     let z be object;
     hereby assume
A1:   z in less_dom (g,r); then
A2:   z in dom g & g.z < r by MESFUNC1:def 11;
      -infty < g.z & g.z < r
        by A1,XREAL_0:def 1,XXREAL_0:12,MESFUNC1:def 11; then
      g.z in ].-infty,r.[;
      hence z in g"(].-infty,r.[) by A2,FUNCT_1:def 7;
     end;
     assume z in g"(].-infty,r.[); then
A3:  z in dom g & g.z in ].-infty,r.[ by FUNCT_1:def 7; then
     ex t be Real st t=g.z & -infty < t & t < r;
     hence z in less_dom (g,r) by A3,MESFUNC1:def 11;
    end;
    hence thesis by TARSKI:2;
end;
