reserve D for set;
reserve x,x0,x1,x2,y,y0,y1,y2,z,z0,z1,z2,r,s,t for Real;
reserve p,a,u,u0 for Element of REAL 3;
reserve n,m,k for Element of NAT;
reserve f,f1,f2,f3,g for PartFunc of REAL 3,REAL;
reserve R,R1,R2 for RestFunc;
reserve L,L1,L2 for LinearFunc;

theorem Th34:
  grad(f,p) = |[ partdiff(f,p,1),partdiff(f,p,2),partdiff(f,p,3) ]|
proof
  grad(f,p) = |[ partdiff(f,p,1)*1,partdiff(f,p,1)*0,partdiff(f,p,1)*0 ]|+
    partdiff(f,p,2)*|[ 0,jj,0 ]|+partdiff(f,p,3)*|[ 0,0,jj ]| by EUCLID_8:59
 .= |[ partdiff(f,p,1),0,0 ]|+|[ partdiff(f,p,2)*0,partdiff(f,p,2)*1,
    partdiff(f,p,2)*0 ]|+partdiff(f,p,3)*|[ 0,0,jj ]| by EUCLID_8:59
 .= |[ partdiff(f,p,1),0,0 ]|+|[ 0,partdiff(f,p,2),0 ]|+
    |[ partdiff(f,p,3)*0,partdiff(f,p,3)*0,partdiff(f,p,3)*1 ]| by EUCLID_8:59
 .= |[ partdiff(f,p,1)+0,0+partdiff(f,p,2),0+0 ]|+|[ 0,0,partdiff(f,p,3) ]|
    by Lm4
 .= |[ partdiff(f,p,1)+0,partdiff(f,p,2)+0,0+partdiff(f,p,3) ]| by Lm4
 .= |[ partdiff(f,p,1),partdiff(f,p,2),partdiff(f,p,3) ]|;
  hence thesis;
end;
