reserve f for PartFunc of REAL-NS 1,REAL-NS 1;
reserve g for PartFunc of REAL,REAL;
reserve x for Point of REAL-NS 1;
reserve y for Real;
reserve m,n for non zero Nat;
reserve i,j for Nat;
reserve f for PartFunc of REAL-NS n,REAL-NS 1;
reserve g for PartFunc of REAL n,REAL;
reserve x for Point of REAL-NS n;
reserve y for Element of REAL n;
reserve X for set;
reserve r for Real;
reserve f,f1,f2 for PartFunc of REAL-NS m,REAL-NS n;
reserve g,g1,g2 for PartFunc of REAL n,REAL;
reserve h for PartFunc of REAL m,REAL n;
reserve x for Point of REAL-NS m;
reserve y for Element of REAL n;
reserve z for Element of REAL m;

theorem Th26:
  (f1+f2)*reproj(i,x) = f1*reproj(i,x)+f2*reproj(i,x) & (f1-f2)*
  reproj(i,x) = f1*reproj(i,x)-f2*reproj(i,x)
proof
A1: dom(reproj(i,x))=the carrier of REAL-NS 1 by FUNCT_2:def 1;
A2: dom(f1+f2) = dom f1 /\ dom f2 by VFUNCT_1:def 1;
  for s be Element of REAL-NS 1 holds s in dom((f1+f2)*reproj(i,x)) iff s
  in dom(f1*reproj(i,x)+f2*reproj(i,x))
  proof
    let s be Element of REAL-NS 1;
    s in dom((f1+f2)*reproj(i,x)) iff reproj(i,x).s in dom f1 /\ dom f2 by A2
,A1,FUNCT_1:11;
    then
    s in dom((f1+f2)*reproj(i,x)) iff reproj(i,x).s in dom f1 & reproj(i,x
    ).s in dom f2 by XBOOLE_0:def 4;
    then
    s in dom((f1+f2)*reproj(i,x)) iff s in dom(f1*reproj(i,x)) & s in dom(
    f2*reproj(i,x)) by A1,FUNCT_1:11;
    then
    s in dom((f1+f2)*reproj(i,x)) iff s in dom(f1*reproj(i,x)) /\ dom(f2*
    reproj(i,x)) by XBOOLE_0:def 4;
    hence thesis by VFUNCT_1:def 1;
  end;
  then
  for s be object holds s in dom((f1+f2)*reproj(i,x)) iff s in dom(f1*reproj
  (i,x) + f2*reproj(i,x));
  then
A3: dom((f1+f2)*reproj(i,x)) = dom(f1*reproj(i,x)+f2*reproj(i,x)) by TARSKI:2;
A4: for z being Element of REAL-NS 1 st z in dom((f1+f2)*reproj(i,x)) holds
  ((f1+f2)*reproj(i,x)).z = (f1*reproj(i,x)+f2*reproj(i,x)).z
  proof
    let z be Element of REAL-NS 1;
    assume
A5: z in dom((f1+f2)*reproj(i,x));
    then
A6: reproj(i,x).z in dom(f1+f2) by FUNCT_1:11;
A7: reproj(i,x).z in dom f1 /\ dom f2 by A2,A5,FUNCT_1:11;
    then
A8: reproj(i,x).z in dom f1 by XBOOLE_0:def 4;
    then
A9: z in dom(f1*reproj(i,x)) by A1,FUNCT_1:11;
A10: reproj(i,x).z in dom f2 by A7,XBOOLE_0:def 4;
    then
A11: z in dom(f2*reproj(i,x)) by A1,FUNCT_1:11;
A12: f2/.(reproj(i,x).z) = f2.(reproj(i,x).z) by A10,PARTFUN1:def 6
      .=(f2*reproj(i,x)).z by A11,FUNCT_1:12
      .=(f2*reproj(i,x))/.z by A11,PARTFUN1:def 6;
A13: f1/.(reproj(i,x).z) = f1.(reproj(i,x).z) by A8,PARTFUN1:def 6
      .=(f1*reproj(i,x)).z by A9,FUNCT_1:12
      .=(f1*reproj(i,x))/.z by A9,PARTFUN1:def 6;
    ((f1+f2)*reproj(i,x)).z = (f1+f2).(reproj(i,x).z) by A5,FUNCT_1:12
      .=(f1+f2)/.(reproj(i,x).z) by A6,PARTFUN1:def 6
      .= f1/.(reproj(i,x).z) +f2/.(reproj(i,x).z) by A6,VFUNCT_1:def 1
      .=(f1*reproj(i,x)+ f2*reproj(i,x))/.z by A3,A5,A13,A12,VFUNCT_1:def 1;
    hence thesis by A3,A5,PARTFUN1:def 6;
  end;
A14: dom(f1-f2) = dom f1 /\ dom f2 by VFUNCT_1:def 2;
  for s be Element of REAL-NS 1 holds s in dom((f1-f2)*reproj(i,x)) iff s
  in dom(f1*reproj(i,x)-f2*reproj(i,x))
  proof
    let s be Element of REAL-NS 1;
    s in dom((f1-f2)*reproj(i,x)) iff reproj(i,x).s in dom f1 /\ dom f2
    by A14,A1,FUNCT_1:11;
    then
    s in dom((f1-f2)*reproj(i,x)) iff reproj(i,x).s in dom f1 & reproj(i,
    x).s in dom f2 by XBOOLE_0:def 4;
    then
    s in dom((f1-f2)*reproj(i,x)) iff s in dom(f1*reproj(i,x)) & s in dom
    (f2*reproj(i,x)) by A1,FUNCT_1:11;
    then
    s in dom((f1-f2)*reproj(i,x)) iff s in dom(f1*reproj(i,x)) /\ dom(f2*
    reproj(i,x)) by XBOOLE_0:def 4;
    hence thesis by VFUNCT_1:def 2;
  end;
  then
  for s be object holds s in dom((f1-f2)*reproj(i,x)) iff s in dom(f1*reproj
  (i,x) - f2*reproj(i,x));
  then
A15: dom((f1-f2)*reproj(i,x)) = dom(f1*reproj(i,x)-f2*reproj(i,x)) by TARSKI:2;
  for z being Element of REAL-NS 1 st z in dom((f1-f2)*reproj(i,x)) holds
  ((f1-f2)*reproj(i,x)).z = (f1*reproj(i,x)-f2*reproj(i,x)).z
  proof
    let z be Element of REAL-NS 1;
    assume
A16: z in dom((f1-f2)*reproj(i,x));
    then
A17: reproj(i,x).z in dom (f1-f2) by FUNCT_1:11;
A18: reproj(i,x).z in dom f1 /\ dom f2 by A14,A16,FUNCT_1:11;
    then
A19: reproj(i,x).z in dom f1 by XBOOLE_0:def 4;
    then
A20: z in dom(f1* reproj(i,x)) by A1,FUNCT_1:11;
A21: reproj(i,x).z in dom f2 by A18,XBOOLE_0:def 4;
    then
A22: z in dom(f2* reproj(i,x)) by A1,FUNCT_1:11;
A23: f2/.(reproj(i,x).z) = f2.(reproj(i,x).z) by A21,PARTFUN1:def 6
      .=(f2*reproj(i,x)).z by A22,FUNCT_1:12
      .=(f2*reproj(i,x))/.z by A22,PARTFUN1:def 6;
A24: f1/.(reproj(i,x).z) = f1.(reproj(i,x).z) by A19,PARTFUN1:def 6
      .=(f1*reproj(i,x)).z by A20,FUNCT_1:12
      .=(f1*reproj(i,x))/.z by A20,PARTFUN1:def 6;
    thus ((f1-f2)*reproj(i,x)).z =(f1-f2).(reproj(i,x).z) by A16,FUNCT_1:12
      .=(f1-f2)/.(reproj(i,x).z) by A17,PARTFUN1:def 6
      .= f1/.(reproj(i,x).z) - f2/.(reproj(i,x).z) by A17,VFUNCT_1:def 2
      .=(f1*reproj(i,x)- f2*reproj(i,x))/.z by A15,A16,A24,A23,VFUNCT_1:def 2
      .=(f1*reproj(i,x)- f2*reproj(i,x)).z by A15,A16,PARTFUN1:def 6;
  end;
  hence thesis by A3,A15,A4,PARTFUN1:5;
end;
