reserve X,Z for set;
reserve x,y,z for object;
reserve A,B,C for Subset of X;

theorem Th60:
  for A being Preorder, f being finite-support Function of A, REAL,
    x being Element of A st
      (for y being Element of A st x =~ y holds x = y) holds
    ((eqSumOf f)*(proj A)).x = f.x
proof
  let A be Preorder, f be finite-support Function of A, REAL;
  let x be Element of A;
  assume A1: for y being Element of A st x =~ y holds x = y;
  per cases;
  suppose A is empty;
    hence thesis;
  end;
  suppose A2: A is non empty;
    then reconsider X = (proj A).x as Element of QuotientOrder(A) by FUNCT_2:5;
    A3: X in the carrier of QuotientOrder(A) by A2, SUBSET_1:def 1;
    A4: x in the carrier of A by A2, SUBSET_1:def 1;
    A5: X = Class(EqRelOf A, x) by Def8;
    for y being object holds y in X iff y = x
    proof
      let y be object;
      hereby
        assume A6: y in X;
        then y in (EqRelOf A).:{x} by A5, RELAT_1:def 16;
        then reconsider z = y as Element of A;
        [x,z] in EqRelOf A by A5, A6, EQREL_1:18;
        then x <= z & z <= x by Def6;
        hence y = x by A1, Def3;
      end;
      thus thesis by A4, A5, EQREL_1:20;
    end;
    then A7: X = {x} by TARSKI:def 1;
    A8: x in dom (proj A) by A4, A2, FUNCT_2:def 1;
    A9: x in dom f by A4, FUNCT_2:def 1;
    per cases;
    suppose x in support f;
      then eqSupport(f, X) = {x} by A7, ZFMISC_1:46;
      then canFS(eqSupport(f, X)) = <* x *> by FINSEQ_1:94;
      then f * canFS(eqSupport(f, X)) = <* f.x *> by A9, FINSEQ_2:34;
      then Sum (f * canFS(eqSupport(f, X))) = f.x by RVSUM_1:73;
      then (eqSumOf f).X = f.x by A3, Def16;
      hence thesis by A8, FUNCT_1:13;
    end;
    suppose A10: not x in support f;
      then not x in eqSupport(f, X) by XBOOLE_0:def 4;
      then {x} <> eqSupport(f, X) by TARSKI:def 1;
      then eqSupport(f, X) = {} by A7, XBOOLE_1:17, ZFMISC_1:33;
      then Sum (f * canFS(eqSupport(f, X))) = 0 by RVSUM_1:72;
      then (eqSumOf f).X = 0 by A3, Def16;
      then (eqSumOf f).((proj A).x) = f.x by A10, PRE_POLY:def 7;
      hence thesis by A8, FUNCT_1:13;
    end;
  end;
end;
