
theorem Th30:
for X,Y be non empty set, f be PartFunc of [:X,Y:],REAL,
x be Element of X, y be Element of Y holds
   ProjPMap1(R_EAL f,x) is PartFunc of Y,REAL
 & ProjPMap1(|.R_EAL f.|,x) is PartFunc of Y,REAL
 & ProjPMap2(R_EAL f,y) is PartFunc of X,REAL
 & ProjPMap2(|.R_EAL f.|,y) is PartFunc of X,REAL
proof
    let X,Y be non empty set, f be PartFunc of [:X,Y:],REAL,
    x be Element of X, y be Element of Y;
    rng(ProjPMap1(R_EAL f,x)) c= rng(R_EAL f) by Th29; then
    rng(ProjPMap1(R_EAL f,x)) c= rng f by MESFUNC5:def 7;
    hence ProjPMap1(R_EAL f,x) is PartFunc of Y,REAL by RELSET_1:6,XBOOLE_1:1;
    rng(ProjPMap1(|.R_EAL f.|,x)) c= rng(|.R_EAL f.|) by Th29; then
    rng(ProjPMap1(|.R_EAL f.|,x)) c= rng(R_EAL abs f) by MESFUNC6:1; then
    rng(ProjPMap1(|.R_EAL f.|,x)) c= rng (abs f) by MESFUNC5:def 7;
    hence ProjPMap1(|.R_EAL f.|,x) is PartFunc of Y,REAL
      by RELSET_1:6,XBOOLE_1:1;
    rng(ProjPMap2(R_EAL f,y)) c= rng(R_EAL f) by Th29; then
    rng(ProjPMap2(R_EAL f,y)) c= rng f by MESFUNC5:def 7;
    hence ProjPMap2(R_EAL f,y) is PartFunc of X,REAL
      by RELSET_1:6,XBOOLE_1:1;
    rng(ProjPMap2(|.R_EAL f.|,y)) c= rng(|.R_EAL f.|) by Th29; then
    rng(ProjPMap2(|.R_EAL f.|,y)) c= rng(R_EAL abs f) by MESFUNC6:1; then
    rng(ProjPMap2(|.R_EAL f.|,y)) c= rng (abs f) by MESFUNC5:def 7;
    hence ProjPMap2(|.R_EAL f.|,y) is PartFunc of X,REAL
      by RELSET_1:6,XBOOLE_1:1;
end;
