 reserve j for set;
 reserve p,r for Real;
 reserve S,T,F for RealNormSpace;
 reserve x0 for Point of S;
 reserve g for PartFunc of S,T;
 reserve c for constant sequence of S;
 reserve R for RestFunc of S,T;
 reserve G for RealNormSpace-Sequence;
 reserve i for Element of dom G;
 reserve f for PartFunc of product G,F;
 reserve x for Element of product G;

theorem Th15:
for G be RealNormSpace-Sequence, p,q be Point of product G,
    r0,p0,q0 be Element of product carr G
  st p=p0 & q=q0
holds p-q = r0
  iff for i be Element of dom G holds r0.i = p0.i - q0.i
proof
   let G be RealNormSpace-Sequence, p,q be Point of product G,
       r0,p0,q0 be Element of product carr G;
   assume A1: p=p0 & q=q0;
   reconsider qq0=(-1)*q as Element of product carr G by Th10;
A2:p-q = p+(-1)*q by RLVECT_1:16;
   hereby assume A3: p-q = r0;
    thus for i be Element of dom G holds r0.i = p0.i - q0.i
    proof
     let i be Element of dom G;
A4:  r0.i = p0.i + qq0.i by Th12,A3,A1,A2;
     qq0.i = (-1)*(q0.i) by A1,Th13;
     hence thesis by A4,RLVECT_1:16;
    end;
   end;
   assume A5: for i be Element of dom G holds r0.i = p0.i - q0.i;
   now let i be Element of dom G;
A6: qq0.i = (-1)*(q0.i) by A1,Th13;
    r0.i = p0.i - q0.i by A5;
    hence r0.i = p0.i +  qq0.i by A6,RLVECT_1:16;
   end;
   hence p-q = r0 by A2,Th12,A1;
end;
